What is the best subwoofer you have ever heard?

Best subs I ever heard were the ones that were properly placed in the room, integrated with the loudspeakers and the associated listening positions, correctly dialed in for volume, phase and correct crossover frequency accordingly, and also inherently well matched with the loudspeakers they're crossing over to!
Bass is THE single most critically setup dependent element when it comes to sound from a system! Place even the most highly touted/rated or expensive subwoofers (self powered, or otherwise) in any system - without successfully addressing all of the afore mentioned points of consideration, and you can forget getting anything resembling exceptional performance from your product!-GUARANTEED! In fact, I'd go so far as to state that the above are all far far more important than the inherent overall performance potential and quality level of the subwoofer itself, yes!
I've dealt hands on with some of the highest end powered consumer woofers ever manufactured (including high end Thiel's, Wilson's, Martin Logan's, Entec, and any number of pro/consumer mid-fi offerings on to M&K,Velodyne,Paradigm, yada yada), and can straight out say that I'd hands down rather have even some modest mid-fi level offerings set up well in any system, over spending big bucks on some potential subwoofer nirvana, that's not set up well! Results are always better with the former..trust me

...Oh, to answer your question (lol), to properly cover all of your stated criteria, I suggest that there's no single subwoofer that can truly do all of that, IMO. Rather, a custom multiple subwoofer project, consisting of large enclosures, many high performance drivers, and driven by lots of power!! All of that, and EQ'd and intricately crossed over to whatever speakers they're mating to!
You want speed, accuracy, ultimate impact, depth, etc. Yep,...custom, expensive, and involved, is what I'm thinking. Otherwise, you'll sacrifice on either end of the performance spectrum, in ultimate terms.
How about a complete front wall covered by a bunch of 10", sealed enclosure, esoteric, hi-end music woofers, powered by like 1000 watts per?!!! Pretty much any distortion would be negated, due to only the singular fundamentals would be audible from the redundant nature of multiple drivers canceling out differences, audibly. And, more importantly, the efficiency of such a setup would only have the drivers working in minimalist output from such an efficient system, to achieve desired output level in relation to the main speakers in the system. Probably gunna be all anyone could ever ask for, for maximum speed and bass impact! Only problem then would be absorbing all that excess bass energy. Heheh...