Which integer sequence starts with small elements, and stays there for a (really) long time, but eventually escapes the initial area?

Question

Graphically, I am searching for something like this:

The only additional requirement would be that the elements are defined by a closed formula or "simple" recursion, i.e. no definition by cases (Fallunterscheidung) and such.

Answer 1

This is not exactly the same thing, but consider the "Tower of Hanoi" sequence:

1 2 1 3 1 2 1 4 1 2 1 3 1 ...

$a_n = k$ when $n$ is an odd number times $2^{k-1}$

We need more than 1000 terms before we see any term greater than 10 (in the Tower of Hanoi puzzle that means if you have 11 layers you need over 1000 moves to expose the bottom disk). Yet the sequence is ultimately unbounded.

Answer 2

How about $a_n=2^{n\cdot\lfloor{n/1000000}\rfloor}$?