I understand that alkaline batteries have about 500mah and what that means in terms of amps and time. But what is the amp limit? Surely they have some limit. I'm wanting to know how many 100mA infrared LEDs I can power with one for a few moments at a time.

 

Well, AudioGoob posted awhile back that he got over an ampere of current from one for a short duration. However, you won't get that level of current for long.

If you want to extend the life of the battery, you can use it to charge a LARGE capacitor via a resistor, and use the cap to power the LEDs for a brief period of time. This will minimize the losses incurred in the battery itself due to internal resistance.

However, you were not specific about the length of time you wanted the LEDs to be on; so it's difficult to say what size cap you'd need.

 

At 9V supply, you can connect several IR LEDs in series, it still consume 100mA.

Exactly how many depends on the LED characteristics and the battery voltage drop when it is loaded.