We could see when we chose our destination folder that there are a lot more raster formats which QGIS can export to via GDAL and other tools. Most of the time, GeoTIFF is sufficient as it can use various compression methods, handle big rasters with the internal BigTIFF format, and use various data types. There are some cases, on the other hand, when GeoTIFF is not an appropriate data exchange format. For example, when I tried to export raster data from IDRISI 7 in GeoTIFF, QGIS could not handle the output. There might have been a problem with IDRISI's GeoTIFF implementation or a bug in QGIS's (GDAL's) GeoTIFF parser--it didn't really matter. GeoTIFF wasn't the appropriate data exchange format for that case and I had to find another one.
From the vast number of raster formats, some of the more widely supported ones are ERDAS's img format and Esri's various binary and ASCII data grid formats. These formats can include some auxiliary files besides the main data file for metadata. For example, the SRTM data we downloaded are in bil files, which is a binary raster format mainly used to store satellite imagery. It is categorized under the ESRI .hdr Labelled group in QGIS, as it comes with an ASCII header file containing some general information about the raster (for example, it uses the BIL format).