It varies from battery to battery, but i have measured the short circuit currents for fresh alkaline batteries:
7.5 A for a 9v battery
8.9A for a AA
AAA I didnt have one to check, but will be less than AA
In general, with the same chemistry, the larger cells provide more current, with D the most and AAA the least
9v is actually 6 AAA cells in series, usually.
NiCd is better than Alkaline cells.
In general not more than a few amps. But it depends on the type of chemicals. From most current to least: lithium, NiMH, NiCd, alkaline, zinc-carbon. It also varies by manufacturer and model. Try to find a technical data sheet to be sure.
9V:
Alkaline: 565 mAh or about half an amp
Zinc-Carbon (aka cheap): 400 mAh or little under half an amp
AA:
Alkaline: 1800-2600 mAh or 1.8 - 2.6 Ah
Zinc-Carbon (aka cheap): 400-1700 mAh or .4 to 1.7 Ah
AAA
Alkaline: 250 – 1200 mAh or .25 - 1.2 Ah
Zinc-Carbon (aka cheap): 540 mAh or .54 Ah
Values vary from brand to brand but those numbers are "typical".
How much current (maximum) can be provided by these batteries?