Chessmaster1000:1st)The probability that both sides will roll a 55 an infinite number of times is exactly zero!

For every move to the infinitum, the probability is 1/36, why would it be smaller at some point?

It's question of possibility not probability.
Ie, if we start counting all the possible games:
....Oh, on this point player 1 can throw 1+1, 1+2 ... 5+5,.. hmm.. let's look more carefully this 5+5. Oh, on this point player 2 can throw 1+1, 1+2, .. 5+5.. hmm.. let's look more carefully this one.. Oh, on this point player 1 can throw ..