How many spins does it take to complete a game of Hi Ho! Cherry-O?
Looking at the work that's been done so far, it seemed to me that this is basically just a Markov chain, and in looking for ways to calculate Markov chains, I found that someone has been there and done that. 

The data from his work matches the simulations here, which should be no surprise given that the rules are very easy to code:


Minimum game length: 3
Expected game length: 15.8
Maximum game length: Unbounded
50th percentile (median): 12
95th percentile: 40
25th percentile: 7
75th percentile: 21



He also posts the Matlab code that he used to calculate the chain ... it's actually pretty simple, comparatively speaking.

The Markov chain transition matrix shown below lists the probabilities of transitioning from one state (i.e. number of cherries pre-spin) to another (number of cherries post-spin).

          # of cherries after spin
       0   1   2   3   4   5   6   7   8   9  10
     --------------------------------------------
  0 | 3/7 1/7 1/7 1/7 1/7  0   0   0   0   0   0
  1 | 3/7  0  1/7 1/7 1/7 1/7  0   0   0   0   0
# 2 | 3/7  0   0  1/7 1/7 1/7 1/7  0   0   0   0
  3 | 1/7 2/7  0   0  1/7 1/7 1/7 1/7  0   0   0
b 4 | 1/7  0  2/7  0   0  1/7 1/7 1/7 1/7  0   0
e 5 | 1/7  0   0  2/7  0   0  1/7 1/7 1/7 1/7  0
f 6 | 1/7  0   0   0  2/7  0   0  1/7 1/7 1/7 1/7
o 7 | 1/7  0   0   0   0  2/7  0   0  1/7 1/7 2/7
r 8 | 1/7  0   0   0   0   0  2/7  0   0  1/7 3/7
e 9 | 1/7  0   0   0   0   0   0  2/7  0   0  3/7
 10 |  0   0   0   0   0   0   0   0   0   0   0