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Arbitrary precision is the correct terminology. I can already represent any size integer I want on any computer. I could just declare that MAXINT represents Graham's number (for example). That doesn't mean that I can represent every integer between zero and Graham's number, though.

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But it's not the precision that's arbitrary it's the range. If your number format represents 0-100 and you want to represent 200 then your number format isn't imprecise, it's limited range.

The "range" is merely established by convention. If your computer supports 100 distinct values then you could say those values represent the integers [0,100) or you could say they represent the even integers in [0,200). The range can be arbitrarily large, but the precision is fixed.

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