I don't want to turn off entire smartparens or similar mode
((([cursor])) When I type ) at cursor location I expect it to be
((())) but it becomes
((()) How to find out which mode is causing it?
According to the description, it looks like elec-pair behavior with user option electric-pair-skip-self set to t.
Firstly, check if electric-pair-mode is enabled. Simply calling M-x electric-pair-mode --which toggles the mode-- should be enough for a quick test.
If the mode is enabled, setting electric-pair-skip-self to nil or #'electric-pair-default-skip-self (default value) will change the behavior to either never omitting a paren or omitting it in accordance with another user option: electric-pair-preserve-balance (start with the default value of t, it should work fine).