1. Bad thermostats (and a lot of stats are solid state now - board and probe)
2. Bad limits (snapped at the capillary tube)
3. Electrical units you'll find broken/burned wiring off elements/contactors.
Experience will be your friend over time. I do HVAC/R and food service stuff, there's been many a time I've worked on a microwave while temping out a reach in, all the while waiting for the building AC to come down after I changed out the compressor earlier, all at the same location. *Note that you're also more valuable, so you'll get paid more.
And I wholeheartedly agree with AtticAce - the worst thing about it is the customers that ***** about the cost, even after our company notifies them in advance the travel and labor rates. I've had checks thrown at me. I've had checks bounce. I had a guy, once I finalized the price, he said he'd write the check when I came back from putting my tools in the truck. He promptly locked his doors when I came back to collect. I had a guy, very effeminate, offer to barter payment, "if you know what I mean" (his exact quote).
Those are rare, though - but fun stories you'll be able to repeat to your grandkids.