"Dirty" compressors: Of course most people don't want any distortion from their compressor, and I usually criticize the distortion I hear from units that are supposed to be clean. But there is a very desirable quality from a pushed/driven tube amp, that includes qualities of both compression and distortion (soft and hard clipping); and there are a few comps that either distort as an accidental side-effect, or that are designed to be able to distort as part of the compression circuit, or that have a separate OD/distortion circuit added on, that can be used for that "amp-like" effect--so here is a list of a few of those units, with brief commentary. For more detailed info, check their individual reviews.
Arion SCO-1: Unintentionally makes nasty lo-fi sounds.
Blackbox Oxygen: Just terrible.
Celmo Sardine Can: Very versatile; absolutely fantastic on guitar, not so great for bass.
dbx 166: It's a very good compressor, and the "limiter" makes a really great distortion effect--but a terrible clean limiter.
DDyna Bass10: Super versatile and unique; its quirky qualities will not suit everyone, but you should try it out.
Durham Sex Drive: Fantastic flavor for all genres and instruments, though it does turn down the lows a bit.
Electro Harmonix Bass Metaphors: Mediocre and disappointing.
Empirical Labs Distressor: Its famous effect is equally loved and hated by pro engineers; good on all instruments, and versatile, but quite expensive.
Fairfield Accountant: Really sweet lively dirty tone. A good natural blend of compression with grind.
Focusrite Bass Factory: The compression and EQ are excellent and extremely versatile; the distortion is OK.
Pigtronix Philosopher's Tone: I wasn't a huge fan of the original version, but the "Germanium gold" one has a very nice smooth distortion that goes well with the heavy sustain. The newer Bass and Rock versions are better (more versatile) as compressors, but their distortion is limited to ON and off, not so useful.
Tech21 Comptortion:Just terrible.
Valley People Dyna-Mite:It doesn't have an actual distortion per se, but it does have a dirty, aggressive quality.
VFE White Horse/Stallion: Very useful, nice-sounding, versatile; good on all instruments, but quirky in its action.
Most digital multi-fx pedals also feature these two effects, but usually the compression and the overdrive/distortion are the two worst sounds in the unit. But who knows, you might find a combo you like in a digital multi. I'll add more devices to this list as I discover and try them.