Power from CR2032 - do I want large cap?

Jiggy-Ninja:
By my math, a 220 uF cap from the Nichicon series would have a maximum leakage current of about 1.3 uA. With just that leakage current loading the battery, a 240 mAh battery would take over 20 years to be deleted.

240 mAh / 1.3 uA = 182,000 hrs = 20.7 years.

Even the worst case leakage gives you a ridiculous lifespan, and depending on how often you're going to be using the remote control might not matter that much. The LED and microcontroller will require many millamps of current, and the energy to send a single IR command might well be equivalent to days worth of sleep leakage, depending on how hard you drive the LED and how well you manage power on the microcontroller when it's active. Depending on how these details shake out it might well turn out that the code sending state has a larger effect on your battery life than sleep leakage. In that case, all your fretting here would be useless (penny wise, pound foolish).

1uA.year=365*24 uAh ~ 10 mAh. In the quoted TI's app note they got 40mAh from adding the cap. If leakage were 1uA any gain is lost in 4 years. For larger leakage it is lost sooner.

One command is 32ms long with 25% duty. With LED current 20mA (->5mA average) and MCU current <1mA sending one command is
32ms*6mA ~ 200uAs ~ 0.05 uAh
Since sending 100 commands should more than be sufficient pressing the button (to get Roomba from its hiding) "costs" about 5uAh. Even 1uA drain from cap would dominate this.

Wawa:
An old tick to measure extreme low cap leakage current is with a DMM set to Volt (not current).

Connect the DMM between a very stable voltage source(+) and cap(+).
The DMM measures voltage across it's own internal 10Megohm (now shunt) resistor.

If you use Ohm's law, you find that 1volt = 100nA leakage.
Charge/equalise the cap first by shorting the DMM leads for a minute.
Cap leakage will drift, so turn the meter off and look again the next day.
Leo..

Nice trick. Will try it, thanks.