I think it depends. My view is that 3 is really the minimum, if you have enough good options, to really have a chance at beating the game without either massive overleveling/grinding or item spamming, both of which I dislike. I really hate having to revive pokemon mid-battle, especially against the Champions. It feels unfair in my mind since they can't do that, and using tons of full restores to PP stall sucks too.
I think it depends both on the challenge and the game you're playing. For example, in my BW monotypes, I try to get a team of 5 if I can for the pregame (5 unova pokemon), which isn't as much of a hardship because of the Lucky Egg, and it ends up being somewhat challenging to fight N and Ghetsis, while also being able to see how the new Pokemon play out. Then I fill out any remaining spots in the postgame with Pokemon from earlier gens that I either like or want to try out. The Lucky Egg is really the only reason I am able to do that without getting bored.
That's just an example, and I play that way because it's fun for me.
If I had to say that there was an absolute minimum, it would be two, since these aren't solos! but three should be the minimum in a perfect world imo.
See, back in the day when I started doing challenges, I would have full teams of 6, like every time (I think my first challenges were the Water RBY monotype and a Challenge where you use Team Rocket's Pokemon.) However, over time, I came to realize that you really didn't need that many Pokemon AT ALL to beat the game. Now, I don't remember when I voluntarily used 5-6 Pokemon for a challenge.
Now, we both know that its not that hard to solo the game with a Pokemon, even with a type disadvantage, a level 80 Pokemon can take out most of the Elite Four with a Body Slam or something. 2 Pokemon is a little tougher, and one might faint from time to time.
I think with 3 Pokemon, a lot of grinding around Victory Road is needed before you take on the Elite Four.
4 and up is just silly imo.
But, I sort of feel dirty just beating the game with 2 Pokemon. It feels like I'm not even trying.
maybe its just because I'm that good at challenges. But, I feel like my main objective is quantity, not quality. So, minimal amount of extra grinding is super fun.
I agree with jd, as you said, 1 or 2 is a bit low for even the most restricted challenges (Except for of course, challenges that is like a duo or solo). So 3 should be a good minimum.
Of course, I can't say for Gen 4 or 5, since I get bored from just playing them since it's so slow without a speed up. But for the first 3 gens, playing with 1 or 2 pokemon is not that hard,
since the pokemon would be overleveled by then, a great deal so. ( Save if you use terrible pokemon of course).
But if you use, 3 pokemon, even if you are stronger than the champ for each pokemon, the elite four would still be able to eat through you if you don't be wary enough.
Let's say the champ's pokemon are at an average 60, that's six pokemon. And your's would be about lv 70~75 at average, maybe even less. But if it were a duo, it would be at and average 60 for the champ and about 80~85 for you! ( I'm not saying that this would mow down the opponents though ) A solo would be even more one sided, the champ's would be 60 and you would be at about 90~ 100! But then again, the gyms would be mowed down, but the champ and the elite four can make life hard for your solo.
So all in all, a 3 pokemon is a good minimum!
I can't speak for all generations, but in Gen I, without excessive grinding, a normal growth rate Pokemon gets you at about 82 at the end of the game, and 2 Pokemon gets you at about 70 each. Gen II is about the same, I think you might be a little lower; about 68 for 2 Pokemon. For 3 Pokemon, I think my last run through Red Version ended with 47, 57, and 62 or something.