I've been running the 3N+1 sequence on a randomly chosen 200,000-bit integer, in hopes that non-4-2-1-terminating numbers become far more frequent at higher values. It's now been running 5.8 days and still hasn't terminated. My implementation is very slow, so I don't know what this means, yet. I unfortunately don't print out iteration numbers as I go, nor do I checkpoint. It is supposed to stop if it iterates 100 times longer than the number of bits it started out with.
It hasn't run out of memory, so the values aren't growing vastly unbounded in size.