I'm going to do it as slowly as I please.
I tried by brute force to at least find a pattern.
Label the paths A, B, C
where A = 2, B = 3, C = 5 hours
and P be the probability of choosing a certain path, and the path ends after traveling path A.
P(A) = 1/3
P(A|B) = P(A|C) = 1/9
P(A|B|B) = P(A|B|C) = P(A|C|B) = P(A|C|C) = 1/27
P(A|B|B|B) = P(A|B|B|C) = P(A|B|C|B) = P(A|C|B|B) = P(A|B|C|C) = P(A|C|B|C) = P(A|C|C|B) = P(A|C|C|C) = 1/81
then the expected value is 2(1/3) + (5+7)(1/9) + (8+10+10+12)(1/27) + (11+13+13+13+15+15+15+17)(1/81) + ... = 2(1/3) + 12(1/9) + 30(1/27) + 112(1/81) + ...
the sum appears to diverge to infinity, but I would have to prove it.