score:14
After going through every Roman Emperor, I found only six who served over a year and did not have a bust or statue in known existence. Here is the list:
My basic methodology for finding this list was
Open every emperor Wiki page who served over one year and hope Chrome doesn't crash
Close every tab with a bust or statue of the emperor present
Google the remaining twenty or so
Advanced search the remaining dozen-ish
And I arrived at these six here.
This is hardly a doctoral thesis, so if anyone finds a bust or statue of one of these men, just remove him from the list and leave a comment explaining where you found the statue/bust.
Upvote:3
I checked on this a while back myself. Aurelian is easily the most significant emperor for whom we do not have a confirmed bust which, like you, I was really disappointed to learn. When you consider he was around for a half a decade, "Restored the World", initiated a great many building projects, and even involved himself in religious matters, it's strange that no bust or statue remains. For now, though, we'll simply have to make due with the portraits we find on his coins.
Otherwise, we have busts even for the majority of emperors whose reign lasted mere months. It's really not until the last string of puppet emperors in the west that a lack of busts becomes the norm rather than the exception... For some, especially Glycerius/Olybrius etc, even their coins are incredibly rare, and as the quality of coins had seriously declined by this point we really have little to no sense of what these late emperors looked like. There may be written descriptions of some of their appearances, but I cannot recall having come across any and, given the sources for that stretch, it's exceedingly unlikely that they'd be very accurate.