Nice! Once this is working, I'd prefer to have this replace BigFloat. Some comments, in no particular order:

It's not completely clear to me what precision MPFRFloat(x::Real) should default to. In many cases, it seems like a better default would be min(DEFAULT_PRECISION, minimum precision to represent x exactly). In this case, a similar default would be used for binary operations '*' etcetera: the min(DEFAULT_PRECISION, min precision to represent result exactly). But maybe this wouldn't be easy to implement.
Would be nice to have a with_precision(p) do ... end construct that temporarily changes DEFAULT_PRECISION within its dynamic scope (similar to a fluid variable in Scheme).
Sometimes specifying the precision in bits and sometimes in digits is a tad confusing.
sqrt should probably throw a DomainError for negative inputs, for consistency with Julia. Similarly for ^.
Would be good to expose more of MPFR's special functions.
convert(::Type{Float32}, x::MPFRFloat) seems missing; not sure if there are other missing conversions.
isnan and isinf seem to be missing
sign (via mpfr_sgn?) seems missing. Also copysign (via mpfr_copysign) and exponent (via mpfr_get_exp) and modf (via mpfr_modf) and rem (via mpfr_remainder) and min/max (via mpfr_min/mpfr_max).
It is incorrect to implement '<=' etcetera in terms of cmp, because this mishandles NaN comparisons. Better to call mpfr_greater_p etcetera.
Might be worth calling mpfr_sum for + with more than 2 arguments (rather than calling the 2-argument mpfr_add repeatedly, as is the default from base/operators.jl).
Would be good to have nextfloat and prevfloat (from mpfr_nextabove and mpfr_nextbelow).
MPFR random-number functions would be good to have access to.
Julia's integer_valued could be implemented from mpfr_integer_p.

switch BigFloat to MPFR & support MPC for complex bignums #2564

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