A prime number random walk
Will we walk arbitrarily far off the y-axis?
Yes, because there are arbitrarily long gaps between consecutive primes.
For every gap $n\in\mathbb{N}$, there is a sequence of $n-1$ consecutive numbers, none of which is prime:
$$n!+2,n!+3,\dots,n!+n$$
In other words, there is no finite bound on the gap between two consecutive primes.
Hence there is no limit as to how far off the y-axis your illustration goes...
Some more graphs:
Up to $n=100000 = 10^5$:
Up to $n=1000000 = 10^6$:
Up to $n=10000000 = 10^7$:
Up to $n=100000000 = 10^8$:
And finally, up to $n=1000000000 = 10^9$: