Double precision floating point numbers are the most common in scientific computing. Sometimes software will mix in single precision for more speed. The molecular dynamics package GROMACS can use mixed single/double precision, for example:

http://manual.gromacs.org/documentation/2016/install-guide/i...

Error analysis can tell you whether the roundoff error would grow exponentially with time steps for the algorithm you use. If you avoid this kind of conditions in scientific computing, then double precision floating point number would work perfectly fine.

And by the way, scientific computing is a diverse discipline, or a federation of disciplines. Say, there may be little overlap between the algorithms, tools, and environments that a physicist uses for scientific computing and those used by a bioinformatician. So I don't think it is universally true in scientific computing that arbitrary precision is preferable to double precision.

I'm quite curious what niche of scientific computing you've dealt with that gave you that impression.