What is the Longest Time I Can Set an Astable 555 Timer?

You are asking for a 10% duty cycle, so the limitation is the 90% of the timing cycle when the output is not on (almost 40 seconds).

Using a CMOS 555 rather than the original bipolar NE555 would be better.

There's not really a hard limit except when the bias current and leakage are such that the oscillator completely fails to work. It will become very inaccurate well before that. The 555 is a 1%-10% device in more-or-less ideal conditions. You have a time shown to five decimal places, when two would be optimistic.

There are special low-leakage electrolytic capacitors, from suppliers such as Nichicon, however they've become more rare in recent decades due to replacement by better digital techniques. Typically you get something like 0.002CV leakage so a 5V 100uF capacitor would have 1uA leakage maximum (roughly...), so 1MΩ would be getting close to not working.

Personally, I'd use an MCU for this, given the limited specifications either with internal oscillator (~1% or a crystal ~20ppm or an oscillator), but there are other digital approaches that replace programmability with multiple parts and soldering.

If your requirement is for an exact 10% duty cycle (fixed and will never change) you could use an astable 555 and a 4017, which means the 555 would create a 0.25Hz pulse train of unimportant duty cycle to drive the 4017. Adding a CMOS divider between the two (such as a 4040) could allow the 555 to operate with a cycle time 20480x higher than the output pulse, so a 120 second output pulse would require the 555 to operate at about 170Hz, which allows the use of a film cap and reasonable-value precision resistors. Also easy to test or calibrate. You'd also need a reset circuit if you care about how it starts up. There are inexpensive ICs for this purpose that do a better and more reliable job than simple circuits such as the execrable R-C-diode network which is (unfortunately) frequently shown.

What is the longest time I can RELIABLY turn my pulse on for. (is 4 seconds the max?

"555" and "reliable" don't tend to mix well, especially as you push the limits of what it can do. That said, things improve if you choose a CMOS 555. With the CMOS device it's possible to use larger resistors. Here's an excerpt from the TI datasheet:

The astable period is set by this equation:

• th = 0.636(Ra + Rb)C
• tl = 0.636(Rb)C
• period (Hz) = th + tl = 1.44 (2Ra + Rb)

This chart would imply that the TLC555 could abide (2Ra + Rb) up to 10 Mohm, compared to 1 Mohm for the bipolar version.

The chart shows about 0.01 KHz (10ms) with (2Ra + Rb) = 1M and C = .15uF. If we scale the cap up to 15uF we get about 10.6 sec. That seems like a reasonable upper bound using ordinary components.

If we try for 10M and 15uF in theory we could get 106 sec. This extreme case would demand a premium low-leakage capacitor as we'll see below when we discuss capacitors.

As the times get even longer, the time accuracy degrades because the rise/fall time is so slow: the charge/discharge trip points will be more sensitive to noise and circuit variations. This will make the timing that much less repeatable. Any RC timer has this issue by the way, not just the 555.

What are good low leakage capacitors (in the tens of micro farad range) that I can use for this project?

Tantalums tend to have lower leakage than electrolytics. In a system I worked on long ago, I saw 22uF tantalums being used for a long-term sample-and-hold. They tended to fail a lot. These days I'd avoid any tantalum caps because they use a conflict mineral (tantalum).

X5R and X7R capacitors are known to have low leakage are are available in that size range as MLCCs. NPO/C0G are lower leakage but tend to be smaller values. Film capacitors are also low leakage, but again may be hard to obtain (and expensive) in larger values. Here's a handy chart showing their relative insulation resistance (IR):

From here: https://passive-components.eu/leakage-current-characteristics-of-capacitors/

X7R looks pretty good and you could easily obtain a 15uF one. Will it work? Let's examine a particular X7R cap here. Its datasheet includes a chart showing insulation resistance (IR) vs. temperature, reproduced here:

The chart is in 'ohm-farads', that is, the insulation resistance is normalized to one farad. Notice that ohm-farads decreases as temperature increases. Over a temp of 20 to 90 deg. C, this X7R cap IR decreases from 1000 down to 100 ohm-farads. Translated to 15uF (that is, [ohm-farads] / [15 x 10^-6 farads]) we see this cap's IR dropping from 67 Mohm down to 6.7 Mohm.

This is ok at normal temp, but at high temp the leakage is a significant considering the proposed largest series resistance (~5 Mohm). As IR decreases, the 'on' time will increase and 'off' will decrease, until the 555 ultimately stops oscillating completely because the leakage never lets the TH voltage reach 2/3 Vcc.

Try a simulation here and see what happens as the leakage goes up.

A film type capacitor would perform much better, offering two orders of magnitude better insulation resistance than X7R, for a price.

Capacitors have other effects, such as dielectric absorption, that influence 'stability' as well. More about that here: https://www.aictech-inc.com/en/valuable-articles/capacitor_foundation05.html

Will the shown circuit be reliable over a long period of time?

Depends on the capacitor. If you're careful about choosing the cap, yes, it could be reliable. The smaller the cap, the less leakage it will have, and the more choices you have for stability, such as being able to choose high-end film caps (see above.)

How can I get a longer pulse? I don't want to use any programmable devices such as microcontrollers.

That's too bad because a microcontroller would give you a low cost and stable solution. Consider the ATTiny for example, available in 8-pin DIP and smaller packages, which could implement the whole solution with no external components. You could have it read a potentiometer or resistor strap to set the time period.

Another option is the Renesas Greenpak, which is best described as a micro-FPGA with some mixed-signal capability. Again, this could implement the entire timer with no external components. Very tiny (maybe too tiny - it's not feasible for most to hand-solder them) but very flexible in what they can do.

Finally, the CD4060 is a CMOS divider with built-in RC oscillator. You can obtain decently-accurate times measured in hours with this device. Example build here: https://www.eleccircuit.com/4060-timer-circuit-project/

Using a '555, one is charging or discharging with a resistor and applying power-supply voltage. The charge/discharge point (the timing capacitor) has a reset transistor connected, with some leakage current.

If, instead of using a '555, one charges or discharges an integrator (op amp with negative capacitive feedback) you can apply, through the charging resistor, plus or minus 10 mV.

Thus, (1) you aren't limited to point-of-charge leakages in the '555, and (2) you aren't limited to power-supply-size applied voltages on the charge and discharge resistance and (3) your triangle-wave thresholds can be anywhere in the op amp output range, not just at one-third V and two-thirds V.

I've made ramp generators with the old CA3140 MOS-input op amp and tens-of-megohms resistors with circa 20 uF capacitance; they could ramp for days before hitting the power supply rails.

Twenty megohms, twenty uF, and +/- 20 mV applied, and toggling at op amp outputs -10V and +10V, makes a triangle/square generator (better triangle linearity than the '555) with period of a couple hundred hours.

One op amp for the integrator, a second op amp as comparator-with-hysteresis to make the square wave.

To get odd duty cycle, offset-adjust at the square-output op amp. Film capacitors are the best (and don't expect 'em to be cheap or compact). Twenty mV on a 20 Mohm resistor is only pushing a nanoamp.

Instead of a '555 for a semiconductor timing circuit, if you want it slow use a dual op amp. Extra credit for good printed wiring board cleanliness.

Here's the Ltspice sim of an option that uses a 555 astable oscillator with a 9.78s period, clocking two CD4017 divide-by-ten counters to give a 97.8s on-time with a 978s period, which has an exact 10% duty-cycle, independent of the clock frequency.

It does use three ICs, but the connections are simple.