What is expected number of turns to play this children's game? I'm playing this game with children and I'm ready to stab my eyes with an ice pick.
It seems like it never ends, but I know I expect it to end. What is my expected number of spins to remove all the fruit from the tree?
Goal: To remove 14 cherries from tree by executing one of following seven directions at random per turn.
 1. Remove 1 cherry.
 2. Remove 2 cherries.
 3. Remove 3 cherries.
 4. Remove 4 cherries.
 5. Return 1 cherry to tree.
 6. Return 2 cherries to tree.
 7. Return all your cherries to tree.

Once I realized I have a 1/7 chance each turn of playing this game in perpetuity, I started reaching for the kitchen drawer.
 A: You can solve via a series of 14 linear equations:
Let $E_n$ be the expected number of turns remaining until the game is over when there are currently $n$ cherries on the tree.
For example, $$E_1=\frac{4}{7}1+\frac{1}{7}(1+E_2)+\frac{1}{7}(1+E_3)+\frac{1}{7}(1+E_{14})$$
$$E_{14}=\frac{1}{7}(1+E_{13})+\frac{1}{7}(1+E_{12})+\frac{1}{7}(1+E_{11})+\frac{1}{7}(1+E_{10})+\frac{3}{7}(1+E_{14})$$
By the time you finish writing down all 14 equations and solving, the game may well be over. (Then again, I expect the answer will be quite large).
A: I've found the reason for the discrepancy between Nate's answer and my results (which agree with DSM's). In the version of the game that Nate linked to, the dog and the bird both require you to return $2$ cherries to the tree, whereas in the present version 5. says one cherry and only 6. says two cherries. If I change my code to the linked version my result for the expected number of turns is in agreement with Nate's. For the present version, I get 
$$\frac{1179248}{80915}\approx14.5739\;.$$
