Approximate the probability An immortal snail is at one end of a perfect rubber band with a length of 1km. Every day, it travels 10cm in a random direction, forwards or backwards on the rubber band. Every night, the rubber band gets stretched uniformly by 1km. As an example, during the first day the snail might advance to x=10cm, then the rubber band gets stretched by a factor of 2, so the snail is now at x=20cm on a rubber band of 2km.

The question: Approximate the probability that it will reach the other side at some point (better approximations are obviously preferred, but any bounds are acceptable as long as they are found by doing something interesting)
With the doubling interpretation, as you have written it, the snail can never reach the end:

Fix the end at $1$, start at $0$

$$X_0=0$$

$$\mbox{suppose} \ \forall n \ X_n=X_{n-1}+2^{1-n} 10^{-5} $$

This is a probability $0$ case, but moving forwards every time is clearly the best case for reaching the end anyway. In this case:

$$\lim_{n \rightarrow \infty}X_n=\frac{2}{10^5}\sum_{n=1}^{\infty} 2^{-n} = \frac{2}{10^5} \ll 1$$
I do not believe the snail will reach the end in finite time, but it may reach the end in the limit, Assuming the extreme case, the snail always moves forward, at time 0, the snail moves 10cm to the right, and is at 10cm on a 1km band. The band then stretches, so the snail is now at 20cm on a 2km band. The snail moves 10cm again and is now at 30cm on a 2km band, but then the band stretches and the snail is at 45 cm on a 3km band.

Let $i$ be a given 'time'. What we have here is that the snail moves $\frac{10}{i\cdot 100,000}$ of the bands length at time $n$, after which the band gets stretched by $\frac{i+1}{i}$, so $\textrm{Pos}(n)$, the snails position at time $n$ (more precisely, before time $n+1$: after movement and expansion) should be:
$$
\begin{align}
\textrm{Pos}(n) &amp;= \sum_{i=1}^n \frac{i+1}{i}\frac{10}{i\cdot 100,000}\\
\textrm{Pos}(n) &amp;= \frac{1}{10,000}\sum_{i=1}^n \frac{i+1}{i^2}
\end{align}
$$
Basically, we have a harmonic series here, which does diverge, but only at infinity, which means that the snail will reach the end with probability 1, but will never reach it at any finite observation.

Once moving backwards is included, it takes on the properties of a 1-dimensional random walk, which, given infinite time, I believe reaches the end, but given finite time won't since even moving forward always doesn't.

Edit
Ross is correct, $\textrm{Pos}(n)$ is the fraction of the band at which the snail sits, so once $H_n$ is greater than 10,000, the snail will hit the end of the band, which is about $e^{10000}$ or a very big number. I am still not sure if it ever hits in finite time when there is positive probability for moving backwards.