Fantastic story. Will read more from this author!
If you run the program once and not observe anything happening, that means you know you're top level. Then you resolve to not rerun the program.
If you run the program once and observe the black sphere, then run it again and not see the black sphere, you know you're level 2. You resolve to not rerun the program. And on it goes.
EDIT: box to sphere.
Imagine you decide to show a "1" instead of the black sphere. You then look behind you. Most likely you see a 1.
Whatever number you see, increment that number by one in the next simulation. If you see nothing, congrats, you're at the top and leave the number as-is.
So initially what each level sees is behind them is: x-1-1-1-1-1
Then after everyone increments by one: x-1-2-2-2-2
Now repeat the step. Everyone looks behind them and increments the number they see by one.
x-1-2-3-3-3
Eventually you'll look behind you and the number won't have changed. You're at one level higher than that.
If there are millions of levels that'd take ages of course. But I think you could use the same idea do it instantly on the computer instead. Just have your simulation computer modify a number on the simulation computer in the next level down.
2. Make the value of n appear in the simulation
3. If you see that number in yours, increment n and go back to Step 2
4. Else, the number you see is how many simulations down you are.