minimum dex count runs

Red/Blue:
Since Squirtle can't learn Cut, the dex count will have to be 2. The options are to either choose Squirtle and catch something else with Cut or choose Bulbasaur or Charmander and catch something with Surf. The best option is likely to choose Bulbasaur/Charmander because there is one very good option for a fully evolved poke that learn Surf: Snorlax. It likely won't be the best as a main for the remainder of the game due to its poor speed and slow level curve but it would definetly be better than having to suffer through with an unevolved starter. The best starter with the Snorlax route will likely be Charmander, as its access to Strength means that Snorlax doesn't need to lose a moveslot for the move.

There's no available poke that can learn both Cut and Fly (Farfetch'd would add two to the dex count instead of one) nor Surf and Fly so the game needs to be done without Fly.

Yellow:
Since Pikachu can't learn Cut, Surf, nor Strength and since there is no poke available before Cut is needed that can learn Cut and Surf, two other pokes will need to be caught for them. As a result, the dex count will have to be 3. Unlike in Red/Blue, Dugtrio can learn Cut in Yellow and there's still Snorlax for Surf. Although it's possible that Snorlax wouldn't need to fight and Dugtrio is good enough at the rest of the game.

There's no available poke that can learn both Surf and Fly so the game needs to be done without Fly

Gold/Silver/Crystal:
Since Totodile can't learn Waterfall nor Strength, the dex count will have to be 2. The best route would involve the red Gyarados as it is the earliest available poke that can learn Waterfall and Strength, as well as just being the best of the remaining options. Red would be quite difficult to get past and Totodile will have to be the starter

There's no available poke that can learn Waterfall, Strength and Fly before Strength is needed so the game needs to be done without Fly

Ruby/Sapphire:
This one's an interesting one. Although Mudkip can learn every move necessary to complete the game, the dex count still needs to be 2 due to the forced Castform acquisition. With that, there a loophole we can exploit to get a good fully evolved pokemon: if we have a full party when we try to get Castform, the scientist will not give us Castform. Therefore, if we catch five of the same fully evolved pokemon (since that won't add to the dex count), we can get a fully evolved poke while still maintaining the dex count at 2. There are a couple of options for this but the best one would likely be Pelipper due to both it being a solid playthrough poke and having access to Fly

Before Pelipper, the run would be absolutely brutal, as an unevolved Mudkip will have a lot of very rough matchups, especially Wattson and Norman

FireRed/LeafGreen:
The route is basically the same as Red/Blue. Snorlax would likely be better with a massively buffed Special Defense stat. Also Dugtrio can learn Cut but it's likely not better than Snorlax, especially as Earthquake is locked until much later

Emerald:
Unlike in Ruby/Sapphire, Castform will always be obtained, even with a full party. This makes the run absolutely brutal as you'd need to truck through the entire game with an unevolved Mudkip and Castform. There's nothing that can be done to get a good fully evolved poke while still keeping the dex count at 1 and to add insult to injury, there's no way to get multiple of the same poke to at least have a full team. Emerald is easily the hardest and there it isn't much of a contest.

Since neither Mudkip nor Castform can learn Fly, the game needs to be done without Fly

Diamond/Pearl/Platinum:
Piplup cannot learn Rock Smash and although Prinplup can, Prinplup cannot have all 5 HMs needed to get through Sinnoh's Victory Road. This means that Turtwig or Chimchar have to be the starter. Afterwards, a poke is needed for Surf and Waterfall and unfortunately there are very few options for good fully evolved pokes that can learn both, since you need to get it before you can Surf. By far the best option is Bibarel if you're allowing X Items but in the case that you ban them, your options are extremely limited. It's essentially just Quagsire which is a fine poke but its poor speed and extreme weakness to grass moves, as well as how it doesn't get Recover until HGSS means that the run would still be a rough one

There is no available poke that can learn Fly, Surf, and Waterfall so the game needs to be done without Fly

Heart Gold/Soul Silver:
Totodile can learn Waterfall unlike in Gold/Silver but it can't learn Strength nor Rock Climb. Thankfully, the best poke available at completing the game, Raikou, can learn both of these moves so after Rival 3, the run becomes essentially the same as the HGSS speedrun in terms of fights

There is no available poke that can learn Fly, Strength, and Rock Climb so the game needs to be done without Fly

Black/White:
The only HM required to complete the game is Cut, which Snivy and Oshawott can learn. However, since the twins on Route 2 will not let you proceed unless you have 2 or more pokes, another poke must be caught before, bringing the dex count up to 2. Unfortunately, there's no way to get a fully evolved poke before the Route 2 twins while still keeping the dex count at 2 so you'd need to struggle through the game with a first stage unevolved poke. This makes Black/White easily the second hardest game for this, however it's leagues easier than Emerald thanks to the introduction of Eviolite and actually being able to catch a full party of the same poke. The best option for this is likely Lillipup due to being the best available poke that can learn Work Up.

There is no available poke before the Route 2 twins that can learn Fly so the game needs to be done without Fly

It's also worth noting that due although the game forces Reshiram/Zekrom to be caught at the end, you can avoid that by filling up *every single PC box*, which you can do while keeping the dex count at 2 by simply catching the same poke every single time. It would take an ungodly amount of time so you better have fun with it.

Black 2/White 2:
Oshawott cannot learn Strength so one more poke would need to be caught. It's also worth noting that the mandatory Charles battle forces at least three pokes in the party. This doesn't mean the dex count needs to be 3, as the three pokes for Charles doesn't need to be different species. The best fully evolved poke available before Charles is Braviary in Route 4 and although it struggles for several reasons, it's still the best option there is. This is especially so because it gives us access to Fly. Thankfully, the Braviary in Route 4 respawns every Monday so you can catch two of them to be eligible to fight Charles

X/Y:
Typical for X/Y, it is the easiest game in this challenge. The free Lucario cannot be avoided meaning that as soon as you get it, you can steamroll the rest of the game. The fights before Lucario (especially before Grant) will likely be a little tricky but nothing a bit of grinding can't accomplish. The Eviolite is available before Korrina (the final pre-Lucario fight) so you can make that fight far easier. The free Kanto starter and forced catch of Xerneas/Yveltal brings the minimum dex count up to 4 and one of the starters you get must be the water one for Surf.

Yveltal can learn Fly but it's only available very late into the game so most of the game needs to be done without Fly

Omega Ruby/Alpha Sapphire:
The free Lati twin (especially if you're playing OR and get Latios) makes the second half of the game (relatively) free, and they even give you access to Fly. It's definitely not as much of a cakewalk as X/Y due to their not great options for STAB and worse matchups into the Elite 4 and Champion but it's nothing a bit of grinding can't accomplish. The pre-lati fights will be absolutely brutal for the same reasons as in RSE, maybe even more so due to having to endure boosted Retaliates from Norman and more powerful STABs from Wattson. Unfortunately, the Eviolite is not available until after Norman.