How many Federation Starships was James T. Kirk responsible for destroying?
Short answer: If you don't count M5's actions, or the class J cargo ship (since Harry Mudd should not have tried to flee), then he destroyed the Constellation, the Reliant, and the Enterprise.
More details: During the series, he destroyed:
- The Constellation in The Doomsday Machine by ramming it down the throat of the planet killer and having it self destruct inside the machine
- In Mudd's Women he puts a tractor beam on Harry Mudd's ship, a class J cargo ship, and Mudd insists on trying to flee, so the ship falls apart. (It could be easily argued that was not Kirk's fault.)
And in The Ultimate Computer, while the M5 was operating, one could say none of these were Kirk's responsibility, but he was technically still Captain of the Enterprise, so they would be considered his responsibility:
- The Woden, a freighter, which may not really be Federation property
- The Excalibur is crippled and all hands are killed by M5. (It's not clear if, by crippled, it could still be towed back and fixed.)
He blamed himself for the death of half the crew of the Farragut, because he hesitated firing the ship's phasers, but he later learned that even if he had fired without hesitating, it would have made no difference.
In the movies he destroyed:
The Reliant in The Wrath of Kahn
The Enterprise (NCC-1701) in The Search for Spock
I'm not even getting into the animated series or non-canon (in other words, what's not on screen). I've gone through an episode listing and reviewed plots for all the episodes and movies. While it's possible I missed something, I think this is pretty complete.