The best way to model this is with a random walk. You start at 0 and then you take a step up to +1 with probability p>.5 and a step down to -1 with a probability q=1-p. You repeatedly take steps either up or down in this manner until you either reach + 1 trillion or - 1 trillion.
There is a formula which says you will reach + 1 trillion first with a probability of [(q/p)^1 trillion-1]/[(q/p)^2 trillion-1], which is very, very close to 1 in the example described.
The formula and its development can be found
here
To answer the question of EVER reaching -1 trillion (ie. going broke), simply consider a repeated application of doubling your money before losing it, first with 1 trillion, then with 2 trillion, then 4 trillion, etc.
Your probability of winning this infinite series of trials is approximated by N*(N^1/2)*(N^1/4)*(N^1/8)*..., where N is your probability of winning your first trial, ie. a number just very slightly less than 1.
This infinite product of probabilities (all less than 1) is equal to
N^(1 + 1/2 + 1/4 + 1/8 + ...) = N^2
which is still very, very close to 1.
So while you are not guaranteed to win after an infinite amount of time, your probability of doing so is very, very high.
The key to understanding why the infinite product of numbers less than 1 can still remain very close to 1 in this case is to understand that the infinite series of fractions 1 + 1/2 + 1/4 + 1/8 + ... equals 2.