I did actually briefly think about which numbers one could construct that way, but then thought that that was not was they were after.
But now that they brought it up, I have a follow-up puzzle:
For which class of numbers in the two intervals is the infinite sequence of signs constructible by a finite machine?
For instance, for pi^2 /6, this is obviously possible: The machine just constructs an infinite stream of "+".
But for most numbers that wouldn't be possible: There are uncountably many numbers in the interval but only countably many machines.