Every time git.exe is executed it returns an exit code (cf. Does git return specific return error codes?). An exit code of 0 indicates success. In the StackOverflow post the exit code 128 is mentioned, which indicagtes an error. The exit code 128 is some kind of "unexpected error" (cf. https://stackoverflow.com/a/8059956/3906760). What it means is highly dependant on the git command executed and the output it issued. Therefore, there can't be a single easy and short answer on how to solve this.

The following post contains a specific answer which is related to the push case and at the end a more generic answer ("If this doesn't help" section).

The following answer is based on the fact, that git.exe was executed to run a push.

Using TortoiseGit with SSH##

Here one needs to differentiate between OpenSSH and PuTTY. (See https://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html#tgit-dug-settings-network)

In order to use OpenSSH you have to set ssh.exe as "SSH client" on TortoiseGit settings -> Network. OpenSSH uses %HOME%/.ssh/id_rsa(.pub) files for their public/private keys by default, which must be created by OpenSSH ssh-keygen and not PuTTYGen (PuTTY keys have a different format, however, can also be converted using PuTTYGen). If %HOME% is not set, TortoiseGit will set it automatically, if you manually set %HOME% please make sure it is valid (you might have to use slashes instead of backslashes).

PuTTY###

PuTTY, which is to be preferred on Windows, requires that public/private keys are created using PuTTYGen (which is shipped with TortoiseGit).  TortoiseGitPlink.exe needs to be selected as ssh client on TortoiseGit settings -> Network - that should be the default. However, there is no default place for the public/private keys and, thus, these have to be configured. There are (at least two ways to do that):

1. Configure your key to be the default key in PuTTY based applications. For this, open PuTTY (which is not shipped with TortoiseGit and has to be downloaded separately). Then go to the Connection->SSH->Auth configuration page and select your .ppk file as "Private key for authentication", go back to the "Session" configuration page, select "Default Settings" there and hit save. -> From now on PuTTY and PLink will use this key for authentication.

2. In order to make TortoiseGit load a private key for using with PuTTY one has to configure it. For an existing repository: Choose a repository, open TortoiseGit settings -> Remote and select your remote repository configuration there (normally "origin"). Then select your private .ppk key and hit "Add new/Save". For a new repository the key can be selected while cloning. When pushing/pulling make sure "Autoload putty key" is checked.

More tips/tricks for using PuTTY: https://tortoisegit.org/docs/tortoisegit/tgit-ssh-howto.html

#If this doesn't help#

First of all we need to make sure that the right git.exe is called - normally that should be the case as TortoiseGit tries hard to find the correct git.exe on the system on first start-up. This can be easily done by going to the TortoiseGit settings -> General. There one should check whether "git.exe Path" points to the right folder of "Git for Windows" (in must point to the bin folder of the Git for Windows installation, not the cmd folder - but this should happen automatically) - if msysgit (the development version of "Git for Windows") is used, it is necessary that the [MSYSGIT-INSTALL-PATH]\mingw\bin-folder is on the path (i.e. entered in the Extern DLL Path textbox) in order to execute git.exe (otherwise required .dll-files) cannot be found. (See https://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html)

Now click on "Check now" and check that the right version number appears next to the button (not no error message). If this is working, git.exe should be configured correctly and work with https.

Every time git.exe is executed it returns an exit code (cf. Does git return specific return error codes?).

An exit code of 0 indicates success.

In the StackOverflow post the exit code 128 is mentioned, which indicates an error:

The (exit code 128) is some kind of "unexpected error" (cf. https://stackoverflow.com/a/8059956/3906760). What it means is highly dependant on the git command executed and the output it issued. Therefore, there can't be a single easy and short answer on how to solve this.

The following post contains a specific answer which is related to the push case and at the end a more generic answer (If this doesn't help section).

The following answer is based on the fact, that git.exe was executed to run a push.

Using TortoiseGit with SSH

Here one needs to differentiate between OpenSSH and PuTTY. (See here)

In order to use OpenSSH you have to set ssh.exe as "SSH client" on TortoiseGit settings -> Network.
OpenSSH uses %HOME%/.ssh/id_rsa(.pub) files for their public/private keys by default, which must be created by OpenSSH ssh-keygen and not PuTTYGen (PuTTY keys have a different format, however, can also be converted using PuTTYGen).
If %HOME% is not set, TortoiseGit will set it automatically, if you manually set %HOME% please make sure it is valid (you might have to use slashes instead of backslashes).

PuTTY

PuTTY, which is to be preferred on Windows, requires that public/private keys are created using PuTTYGen (which is shipped with TortoiseGit). 
TortoiseGitPlink.exe needs to be selected as ssh client on TortoiseGit settings -> Network - that should be the default. However, there is no default place for the public/private keys and, thus, these have to be configured. There are (at least two ways to do that):

1. Configure your key to be the default key in PuTTY based applications. For this, open PuTTY (which is not shipped with TortoiseGit and has to be downloaded separately). Then go to the Connection-> SSH -> Auth configuration page and select your .ppk file as "Private key for authentication", go back to the "Session" configuration page, select "Default Settings" there and hit save. -> From now on PuTTY and PLink will use this key for authentication.

2. In order to make TortoiseGit load a private key for using with PuTTY one has to configure it.

• For an existing repository: Choose a repository, open TortoiseGit settings -> Remote and select your remote repository configuration there (normally "origin"). Then select your private .ppk key and hit "Add new/Save".
• For a new repository the key can be selected while cloning. When pushing/pulling make sure "Autoload putty key" is checked.

More tips/tricks for using PuTTY.

If this doesn't help

First of all we need to make sure that the right git.exe is called - normally that should be the case as TortoiseGit tries hard to find the correct git.exe on the system on first start-up. This can be easily done by going to the TortoiseGit settings -> General. There one should check whether "git.exe Path" points to the right folder of "Git for Windows" (in must point to the bin folder of the Git for Windows installation, not the cmd folder - but this should happen automatically) - if msysgit (the development version of "Git for Windows") is used, it is necessary that the [MSYSGIT-INSTALL-PATH]\mingw\bin-folder is on the path (i.e. entered in the Extern DLL Path textbox) in order to execute git.exe (otherwise required .dll-files) cannot be found. (See TortoiseGit docs)

Now click on "Check now" and check that the right version number appears next to the button (not no error message).
If this is working, git.exe should be configured correctly and work with https.

Generic answer

Every time git.exe is executed it returns an exit code (cf. Does git return specific return error codes?). An exit code of 0 indicates success. In the StackOverflow post the exit code 128 is mentioned, which indicagtes an error. The exit code 128 is some kind of "unexpected error" (cf. https://stackoverflow.com/a/8059956/3906760). What it means is highly dependant on the git command executed and the output it issued. Therefore, there can't be a single easy and short answer on how to solve this.

The following post contains a specific answer which is related to the push case and at the end a more generic answer ("If this doesn't help" section).

Answer for the case of the initial Stackoverflow post

The following answer is based on the fact, that git.exe was executed to run a push.

Here one needs to differentiate between OpenSSH and PuTTY. (See http://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html#tgit-dug-settings-network)

In order to use OpenSSH you have to set ssh.exe as "SSH client" on TortoiseGit settings -> Network. OpenSSH uses %HOME%/.ssh/id_rsa(.pub) files for their public/private keys by default, which must be created by OpenSSH ssh-keygen and not PuTTYGen (PuTTY keys have a different format, however, can also be converted).

1. Configure your key to be the default key in PuTTY based applications. For this, open PuTTY (which is not shipped with TortoiseGit and has to be downloaded separately). Then go to the Connection->SSH->Auth configuration page and select your .ppk file as "Private key for authentication", go back to the "Session" configuration page, select "Default Settings" there and hit save. -> From now on PuTTY and PLink will use this key for authentication.

2. In order to make TortoiseGit load a private key for using with PuTTY one has to configure it. For an existing repository: Choose a repository, open TortoiseGit settings -> Remote and select your remote repository configuration there (normally "origin"). Then select your private .ppk key and hit "Add new/Save". For a new repository the key can be selected while cloning. When pushing/pulling make sure "Autoload putty key" is checked.

First of all we need to make sure that the right git.exe is called - normally that should be the case as TortoiseGit tries hard to find the correct git.exe on the system on first start-up. This can be easily done by going to the TortoiseGit settings -> General. There one should check whether "git.exe Path" points to the right folder of "Git for Windows" (in must point to the bin folder of the Git for Windows installation, not the cmd folder - but this should happen automatically) - if msysgit (the development version of "Git for Windows") is used, it is necessary that the [MSYSGIT-INSTALL-PATH]\mingw\bin-folder is on the path (i.e. entered in the Extern DLL Path textbox) in order to execute git.exe (otherwise required .dll-files) cannot be found. (See http://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html)

Here one needs to differentiate between OpenSSH and PuTTY. (See https://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html#tgit-dug-settings-network)

In order to use OpenSSH you have to set ssh.exe as "SSH client" on TortoiseGit settings -> Network. OpenSSH uses %HOME%/.ssh/id_rsa(.pub) files for their public/private keys by default, which must be created by OpenSSH ssh-keygen and not PuTTYGen (PuTTY keys have a different format, however, can also be converted using PuTTYGen). If %HOME% is not set, TortoiseGit will set it automatically, if you manually set %HOME% please make sure it is valid (you might have to use slashes instead of backslashes).

1. Configure your key to be the default key in PuTTY based applications. For this, open PuTTY (which is not shipped with TortoiseGit and has to be downloaded separately). Then go to the Connection->SSH->Auth configuration page and select your .ppk file as "Private key for authentication", go back to the "Session" configuration page, select "Default Settings" there and hit save. -> From now on PuTTY and PLink will use this key for authentication.

2. In order to make TortoiseGit load a private key for using with PuTTY one has to configure it. For an existing repository: Choose a repository, open TortoiseGit settings -> Remote and select your remote repository configuration there (normally "origin"). Then select your private .ppk key and hit "Add new/Save". For a new repository the key can be selected while cloning. When pushing/pulling make sure "Autoload putty key" is checked.

First of all we need to make sure that the right git.exe is called - normally that should be the case as TortoiseGit tries hard to find the correct git.exe on the system on first start-up. This can be easily done by going to the TortoiseGit settings -> General. There one should check whether "git.exe Path" points to the right folder of "Git for Windows" (in must point to the bin folder of the Git for Windows installation, not the cmd folder - but this should happen automatically) - if msysgit (the development version of "Git for Windows") is used, it is necessary that the [MSYSGIT-INSTALL-PATH]\mingw\bin-folder is on the path (i.e. entered in the Extern DLL Path textbox) in order to execute git.exe (otherwise required .dll-files) cannot be found. (See https://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html)