Are there any algorithms so slow that big O notation cannot bound them?

Question

Is there an algorithm (that is guaranteed to halt) that has such great time complexity that it cannot be bounded asymptotically by a computable function? I understand the busy beaver function BB(x) grows faster than any computable function, but I also think that no algorithm could be designed to run in Θ(BB(x)) as this would solve the halting problem.

I think the answer is no, but I'm not sure how to prove this.

Answer 1

An algorithm that always halts has a running time that is computable by definition: imagine running the program in an emulator and tallying up the cycles spent by the program.