PLY format reuses vertices, so it does not suffer the issues of STL. I for one do single out STL as the worst popular mesh format as it is the only one that does not reuse vertices.
As @lassoan notes, a major disadvantage of OBJ is that it is only ASCII. Both PLY and STL have ASCII and binary variants. With ASCII you need to trade off file size and precision, and file loading is always extremely slow. In my software OBJ (with moderate precision) is about 13 times slower and 7 times larger than a simple gzipped binary format (MZ3).
The MZ3 format can be extended, is very fast and relatively compact. I might be biased here as I developed it, but I would be happy to extend this. The current version was extended to support features for PALM (Anderson Winkler) and SPM (Guillaume Flandin) and I provide Matlab, JavasScript and Python/Blender code. The strength of this format is that is virtually identical to the raw data one sends to the GPU, which makes it fast. However, just like modern GPUs it only represents triangles. The advantage of triangles is that as long as three points are not co-linear they define a single plane, while this is not true of all quads (e.g. all 3-legged stools are stable, but many 4 legged chairs rock between legs). While modern GPUs do not use quads as a base type, quads can be useful for a file format to store if one is interested in subdivision.
The GIfTI format does allow meta data. It is XML-based and so it is human readable. It uses base-64 storage for binary data which adds a bit of a speed and file size hit, but this is pretty modest. It seems like a reasonable interchange format for brain imaging.
The Draco format is very clever, creating extremely compact files. However, it is very complicated and the fact that it re-indexes vertices would probably be a problem for a lot of brain imaging applications.