I'd like to explain why I don't think we've really seen an exploration of a true jack-of-all-trades.
Mew is the closest thing, yes - but, while it can do just about everything, it's not effective enough at a lot of those jobs for it to be worthwhile, largely due to its mediocre typing. Having Psychic for its only STAB is a major blow to Mew's offensive capabilities, especially when its base 100 offenses are at the lower bound of usable. Pokémon with similar offensive stats (for instance, Infernape or Victini) rely on high-power moves from great STAB types to be real offensive threats, which Mew can't do - hurting its ability to truly fill any role that's desired.
Mew can run a Swords Dance set, but it's all-around less effective than Nasty Plot as a simple wallbreaker; Zen Headbutt gives you similarly mediocre power with the addition of slightly questionable accuracy, and your coverage options aren't nearly as good. Choice Specs or Choice Band just don't do enough damage; your power is an issue even with Nasty Plot. Forget about a Choice Scarf set, for much the same reason. And a dedicated defensive set? Psychic typing is... problematic. Yeah, it gives you a Fighting resistance, which is neat - but not at the cost of being weak to Pursuit and the ubiquitous U-turn. Mew's best bet defensively is the support set on site right now; that's good at its job, mind you, but it can't really stay in and wall a lot of dangerous threats. Will-O-Wisp is the best thing it has going for it, so Special offense can tear through Mew and its lacking set of resistances, while faster U-turners can take a huge chunk out of its HP. Mew has unlimited options, but it struggles to exploit a lot of them. The goal of this concept would be to make a Pokémon that can fully utilize every tool we give it - becoming a threat because of sheer unpredictability and the element of surprise. I'd imagine lure sets would be very common; if the common physical tank is checked by CB Terrakion, you might run a physically offensive Scarf variant with Close Combat on what is otherwise a more defensive team that would be expected to run the tank, if that team has problems with Terrakion. Terrakion comes in and is cleanly 2HKO'd; you retain a serviceable revenge killer and have eliminated a major threat to your team as a whole.
It would be very hard to come closer to the concept than Mew, mind you; you're walking a fine line, making a Pokémon that can do everything, can do it well enough to be dangerous in that role, but isn't overpowered. That being said - I think that we could pull it off, as long as we were cautious. The importance of a good typing to this concept really can't be overstated; the wrong typing will shut the CAP out of potential roles regardless of its stats and movepool.