I think you can develop an algorithm that gives all integers (without limit) SOME probability of happening. You just can't develop an algorithm that (a) terminates, and (b) gives them all equal probability.
I can try to give you an example of "an algorithm that gives all integers (without limit) SOME probability of happening" if you like.
edit. I'll just give it now, I just thought of one.
The algorithm is to produce a number in binary. Every step involves 2 flips.
flip 1: determine if the digit you're on should be a 0 or a 1, heads is 1, tails is 0.
flip 2: determine if you should STOP or generate another digit - heads means generate another digit, tails means stop.
so your first flip, you do Heads, so
1.
your next flip, you do Heads, so flip another.
your next flip, you do Tails, so
01.
your next flip, you do Heads, so flip another.
next flip is heads, so
101.
next flip is Tails, so stop.
final number in binary is 101, converted from binary to decimal is
5
This algorithm is theoretically capable of randomly producing ANY of the infinite sequence of integers, but it preferentially chooses lower numbers.