What are some franchises where you think the second game is the best?

To me, the posterchild for this has to be Star Fox 64. They perfected the formula here, seemingly so well that they decided never to revisit it because they know it cant be topped. I dont agree with that philosophy and i think it could be topped, especially with the hardware we have these days, but you onow how Miyamoto can be. 

Another one that i dont think many would disagree with is Uncharted 2. My personal preference might be the first, but if i put my objective cap on, O have to admit the second did improve on basically every aspect. 

The last one ill go with maybe doesnt count, since its not really the second donkey kong game, but its the second in the mainline platformers that we have all come to expect from a donkey kong game, and thats Donkey Kong Country 2. This is another game that has taken everything from the first and improved on it in every imagineable way. To me its the clear winner of the trilogy and it beats out every DK game we have had since, including DK 64, the returns games, and bananza. 

Bayo 2, best action game created.

Tomb Raider II
Tony Hawk's Pro Skater 2
I go back and forth on Doom and Doom II
And probably some iteration of Street Fighter II

SSX Tricky
Age of Empires 2
Rollercoaster Tycoon 2

SSX Tricky is one of the greatest games ever crafted. Thank you for your inclusion of this game on your list. I would love to see that game on NSOnline one day… though I’m a tad bit doubtful that Nintendo is gonna go out of their way to reach a deal with EA.

Responding to OP: Majora’s Mask, Sonic Adventure 2, and (maybe a tad bit controversially) Tears of the Kingdom. One could also make a strong case for Tropical Freeze.

• StarCraft 2
• Portal 2
• Pokémon Gold and Silver
• Half Life 2 (sort of, if you don't kind Black Mesa)
• Borderlands 2
• Alan Wake II
• System Shock 2
• Ori and the Will of the Wisps

Penguin Adventure
The Maze of Galious
King's Valley 2

Sonic 2
Street Fighter 2 (not big on the series though)
Mortal Kombat 2 (at least in wow factor and atmosphere)
Streets of Rage 2
Suikoden 2 (I hardly played the other games in the series but I don't see them topping this).
Dino Crisis 2
Time Crisis 2
Rival Schools 2 (Project Justice)
Zone of the Enders 2
Jak 2 (arguably)
Little Nightmares 2

Uncharted 2

Onimusha 2 comes close but I prefer Onimusha 3's combat, and I really enjoyed Ako's presence. And Samanosuke is the best.

Maybe Gradius 2? (Well...  Gradius 2 "Gofer no Yabou" is technically the 3rd Gradius. Nemesis 2 on MSX is the original Gradius 2).

Will add more as I remember them or someone reminds me of them.

WarCraft 2, foe sure.

SOCOM II
Wasteland 2

Danganronpa 2. It's for sure the best in the series in my view and a contender for best VN of all time.