People believe (although there is no proof) that any finite sequence of bits appears somewhere in pi, making pi an illegal number :)
https://math.stackexchange.com/questions/216343/does-pi-cont...
https://math.stackexchange.com/questions/216343/does-pi-cont...
The other idea I've always liked is a length that, to enough precision, encodes something in the decimal places. Then in theory you could just cut something, say a diamond, to this impossible precision, and then measure it later. I assume that, like the Pi example, this is impractical not only in practice but even in theory, even if we had impossibly-accurate cutters and measurers.