[kens]

Looking historically, you have a bunch of options for a pre-IC computer; there were lots of pre-IC computers. Transistors, of course, or vacuum tubes give you a useful computer. You can build a computer from relays, but the performance is pretty bad. Memory is also very important. Magnetic core memory is the way to go if you don't have ICs. None of this is going to help you if you went to the dark ages.

As far as mechanical devices, mechanical calculating machines didn't arise until the late 1600s and weren't reliable for many years. It's unlikely that you'd be capable of building a mechanical computer until the industrial revolution. Note that Babbage was unsuccessful in building his machines even in the late 1800s.

If your goal is to build a Turing-complete machine of some sort, even if totally impractical, you could push the date back a lot. But that would be more of a curiosity than a useful computer.

[Joker_vD]

For arithmetic, pinwheel calculator (aka "Odhner's arithmometer") [0] is a pretty decent and reliable mechanical device. You can even give it an electric motor for doing the rotations for you and a numerical keyboard.

[0] https://en.wikipedia.org/wiki/Pinwheel_calculator