I have never seen anything of the sort ;). Something that is of considerable interest in the data science community is half-precision reals, because they have little to gain with six decimals but everything with less memory per number. That would, however, still fit in a more or less ordered sequence of number formats - the precision is reduced along with the range. Your example would upset that.
I have seen a Convex machine in years long gone by where there were two number formats. The main difference being a different interpretation of the exponent encoding and a slightly faster processing with the native number format (that was not IEEE-compliant). But that was before Fortran 90 was a serious option. You had to select the format via a compiler option.
Ads it happens there is a thread on the gfortran mailing list that concerns the handling of signalling NaNs. A large number of platforms are mentioned there each with its own ideas about what a number should be (*). Perhaps one of them has something akin to the number format you mention.
(*) Is it my too poor mastering of the English language that I have to use the word “number” in at least two different meanings?