Yeah, the first two subs I think of, slightly over budget, slightly too large, are
aforementioned SVS: Approx. 15” deep x 14" wide x 14 "
Epik Vanquish: Size: 14.5 H x 14.5 W x 14.5 D (15.5 w/ grill)
Something that is closer to that super tiny size you are looking for, and a popular choice, is the Martin Logan Dynamo: 13.9" × 11.7" × 12.9". However, it really does give up too much down low, IMO, but then again I spent a lot on mine.
Sunfire also makes a tiny sub, but, at these sizes, as with the Dynamo, it's just not really possible... save annunaki's idea. JL makes a 10" sealed that should be formidable, but it blows the budget sky high, and it STILL is too large for you at 15.64 in x 12.92 in. x 17.27 in. OTOH, I would presume it would be the best performing out of the listed commercial subs. Of course, you pay for that though.