How many AES-operations have to be watched for key-exploit by side-channel attacks?
As pointed out by poncho, it is not possible to precisely answer your question since too many parameters are involved:


About the AES implementation: is it an unsecured implementation? Does it use masking? If yes, at which order? And so on...
The physical properties of the device also matter, does it leak a lot? Are there some hardware countermeasures such as a shield, a jitter, etc...
What equipment is used to measure leakages? Depending on the probe, the oscilloscope and its setup, the number of AES executions to achieve a 100% success rate may vary greatly.


To put it in a nutshell, you have to make practical experiments to answer your question.