Solved by Paul Brinkley

solution: heaven

1/3 of the time, the door to heaven will be chosen, so 1/3 of the time it will take zero days. 1/3 of the time, the 1-day door is chosen; of those, the right door will be chosen the next day, so 1/9 trips take 1 day. Similarly, 1/9 will take two days (choosing the 2-day door, then the right door).

After that, the cases split again, and again, and again. I can't seem to make a nice infinite sum this way, so let's try again.

Suppose the average days spent is X. 1/3 of the cases are done in zero days as before. 1/3 of the cases are 1 day plus X. 1/3 are 2 + X. So:

X = 1/3 * 0 + 1/3 * (1 + X) + 1/3 * (2 + X)
  = 0 + 1/3 + X/3 + 2/3 + X/3
  = 1 + 2X/3

Therefore,

  X/3 = 1
    X = 3

On average, it takes three days to get to heaven. Two if the noodles are limp.

Took me one blind alley, and about five minutes. (heh heh)