@DLosc Would you know, I just proved that a few weeks ago. esolangs.org/wiki/Brainfuck#Computational_class

(Just two levels of nesting.)

Although looking at Acc!!, I suspect something like Fractran or ais523's Tip might be easier targets.

@ØrjanJohansen Cool! Does that use a tape that is bounded to the left or unbounded in both directions? (I should be able to make the proof work either way, but bounded to the left might make it easier.)

@ØrjanJohansen I don't see a good way to store lists of arbitrary rational numbers in Acc!!. Generally, I've stuck to using the accumulator for lists of bounded nonnegative integers (plus up to one unbounded integer). But there could certainly be a strategy I'm missing.

@DLosc It's bounded to the left.

@DLosc The strategy I'm envisioning is compiling to Acc!!, so the rational numbers for the program don't need to be stored simultaneously in the accumulator.

OTOH the brainfuck construction only needs a tape of bits, so you might still find that easier if you know how to handle that.