This is ready for review. I would like to merge this in few days to use that for mpf.__format__().

How would you use this functionality with mpf rather than _mpf_?

>>> import mpmath
>>> from mpmath.libmp import to_str, from_str
>>> f = mpmath.sqrt(2)
>>> to_str(f, 10, base=2)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/oscar/current/active/mpmath/mpmath/libmp/libmpf.py", line 1128, in to_str
    if not s[1]:
           ~^^^
TypeError: 'mpf' object is not subscriptable
>>> to_str(f._mpf_, 10, base=2)
'0b1.0110101'

It looks like the mpf constructor accepts it:

>>> mpmath.mpf('0b1.0110101')
mpf('1.4140625')

How would you use this functionality with mpf rather than mpf?

First, we could provide __format__() dunders to support "new-style" formatting of mpf/mpc's (issue #337). See e.g. gmpy2: https://gmpy2.readthedocs.io/en/latest/mpfr.html#gmpy2.mpfr.__format__ I'm planning to do this like python/cpython#113805 (i.e. "x" will be used to format numbers in the hexadecimal notation). Another option (as we don't support old-style string formatting) may be using "a" letter for this (in printf-style, like the MPFR). Support for optional rounding mode (gmpy2-like) also does make sense for me (see #757). Maybe we should also support formatting binary floats ("b"?).

It looks like the mpf constructor accepts it

Yes, strings with prefixes should be accepted automatically. Most important case is hexadecimal floats, of course - this will allow us to pass exact floating point values to the mpf constructor (see also CPython pr: python/cpython#114668).

But to handle arbitrary bases we should add base keyword argument (default=0) to mpf/mpc constructors (as gmpy2 does).