I actually spent some time about a year ago doing some computations for a variant of this game, sold as Hi-Ho Cherry-O. It's identical to your game, except with 10 cherries instead of 14. (I learned about it from a colleague with a 4-year-old daughter.)
The computation is a nice example of some simple Markov chain techniques, which produce linear equations of the sort in Brett Frankel's answer. I considered the cases of 1 to 4 players, which are amenable to computer solution.
Another interesting feature is that since the players take turns, the first player has a slight advantage.
Here are the results I got for 10 cherries. If you are really interested, I can try and reconstruct my code and run the 14 cherry case.
1 player game:
Expected length: 15.8019792994073 rounds
2 player game:
Expected number of rounds: 9.58554137805221
P(player 1 wins) = 0.518720469382215
P(player 2 wins) = 0.481279530617784
Expected number of turns = 18.6523622867222
3 player game:
Expected number of rounds: 7.49668096168849
P(player 1 wins) = 0.357756582790784
P(player 2 wins) = 0.332728455615310
P(player 3 wins) = 0.309514961593905
Expected number of turns: 21.4418012638686
4 player game:
Expected number of rounds: 6.44149249272987
P(player 1 wins) = 0.276928283784381
P(player 2 wins) = 0.258099951775544
P(player 3 wins) = 0.240610168544412
P(player 4 wins) = 0.224361595895655
Expected number of turns: 24.1783750474708
Edit: I should also mention some previous work by Jeffrey Humpherys.