I don't think the rule about sequels really exists in games. It's more of a movie thing. A ton of games have sequels that are better than the original. Like... a ton.

If you want a specific example of the third one being the best though, my favorite game of all time is The Elder Scrolls III: Morrowind, so there ya go. ;)

Kings Field II, which was the third game in the franchise, is the only one I can think of that hasnt been named.
Oh, and Dark Souls II, if you count Demon's Souls as the first one, which I do.

The other stand out for me is Morrowind, which is amazing.

I second this.

Worms: Armageddon is the third Worms game.

I think that's more of a rule in movie business, where franchises are milked to death. Some famous director makes a movie which is a huge success, then walks away to create something new and different. Because of the success some less famous director is tasked to make a sequel which is essentially "more of the same", because of the name it becomes a mild success, which again warrants a low-budget direct-to-video "threequel" by whatshisname...

Games work differently, although, of course stuff like that also happens, and milking of franchises is also a thing. But often sequels are by the same guys who made the original game, which gives them the opportunity and budget to finally implement all those ideas and features they had for the first game but couldn't implement due to time and money constraints, and also improve on the things that didn't work so well before - or generally go for bigger, better, more badass.
And with part 3 it really depends if they still have enough stuff to improve, new directions to go into, or if it's just a "we do it for the money" thing. And even with the latter - some series don't fare so badly with numerous re-iterations which update visuals and UI and not much more. The same game all over again with better graphics is still a "better" game for most people.

Police Quest 3, Quest for Glory III and Gabriel Knight 3 are the best games in their respective series as well.

I really struggled with RCT3. I didn't feel like it had the charm of RCT2 (don't get me wrong, it's a fine technical achievement and the graphics are nice, but I felt that the look and feel of RCT2 beat 3 hands down).

Perhaps I need to give it another go.

Some more games, although some of them are worse storywise, or in terms of feeling/atmosphere. However, I still feel they improve on the games before in other areas.

God of War 3 - The story was weaker, Kratos was trully unlikable but this game was massive compared to the previous two. The scale, the bosses, the graphics, the gameplay. One of the best action games out there in this category.

Resistance 3 - A more personal story. The stakes were higher. Like the first one, you were able to carry all weapons. The levels were more interesting too.

Max Payne 3 - Totally different atmosphere than the previous two. Some say that it doesn't feel like a Max Payne game. But this one is so good. The shooting is top notch.

BioShock Infinite - Yeah, the controversial one. Sure the twists and turns can be hard to take in, the game is not what was promised, but I still think this is a very solid game. Add in the two dlc set in Rapture and it makes a very good package.

Crysis 3 - I think they made a good jog of balancing the more expanded design areas of the first one with the verticality and corridor type levels of the second. The addition of the bow, that didn't cloaked you off when shooting it, was very nice and enabled a more stealthy approach to some situations. I also like the ending.

Caesar III, Might and Magic III the isles of Terra (I find the 1 and 2 little unplayable), Super Mario bros 3, agree whit halo 3 :)

Super Metroid (SNES, 1994)
Super Mario Bros. 3 (NES, 1988/SNES, 1993)
Mother 3 (GBA, 2006)
Wonder Boy III: The Dragon's Trap/Monster World II (SMS, 1989)
The Legend of Zelda: A Link to the Past (SNES, 1991)
Heroes of Might and Magic III: The Restoration of Erathia/Heroes 3 (PC, 1999)
Wario Land: Super Mario Land 3 (GB, 1994)
Super Bomberman 3 & Bomberman '94/Mega Bomberman
Sonic the Hedgehog 3 & Knuckles (MD)
Mega Man 3 (NES, 1990/PS1)
MUSHA (Aleste 3 basically)
Warcraft III
Thunder Force III
Micro Machines: Turbo Tournament '96 (MD)
Super Fantasy Zone
Shinobi III: Return of the Ninja Master (MD)
Shining Force III
Ganbare Goemon 3
Castlevania: Aria of Sorrow (at least I think this is considered the third after SotN)
Mega Man Zero 3 (GBA)
Darius Gaiden (ARC, 1994/SAT, 1995)
Tekken 3 (PS1, 1998)
F-Zero GX
Teenage Mutant Ninja Turtles III: The Manhattan Project/TMNT III (NES)
Adventures of Lolo 3 (NES)
Castlevania III
Populous: The Beginning (PC)
Demon's Crest (SNES)
Bare Knuckle 3/Streets of Rage 3 JP (MD)
Dragon Quest III
Mario Kart Wii
Street Fighter Alpha 3
Final Fantasy III (NES)
King's Quest III (PC)(AGD Remake)
Splatterhouse 3
Final Fight 3 (SNES)
Caesar III

Actually, I believe that's actually the 4th game in the series; Wonder Boy 3: Monster Lair came out in 1988.

With that said, Dragon's Trap is the first game in the series that is anything like any previous entry in the series, which is rather strange.

(Reminds me of Touhou, where Touhou 4 is the first entry to be similar to an earlier game in the series; 1 and 3 are not typical of the series (3 is like 9 and 1 is unique).)

Strongly agree with Saints Row The Third, disagree with Curse of Monkey Island (it has its moments, but it also redundantly rehashes things that have been done better in the previous games, just for the fan service; the first two games were more original).

A bit more controversial, but I also like Far Cry 3+ more than what came before it.

I also think StingingVelvet and toxicTom have a point. I'm not even sure there is such a thing as a "rule" with exceptions regarding movies. A lot of sequels suck because the first installment was fresh and original while the sequel just tries to cash in on that without innovating anymore; or because the innovations are ill-advised and rob the game of its original charm - it's a tricky balance. But sequels can also be better if they manage to capture what was good about the original and refine it. In some cases the first installments need to invest quite a bit of time on ouvertures and introduction, which can lead to a sequel being more enjoyable because you're already familiar with setting and characters now and that allows for more interesting subplots and character development (e.g. X2: X-Men United, Spiderman 2, Harry Potter and the Chamber of Secrets - IMO, I know many people are not so fond of it).

Galciv III

Actually not really, it depends what you count as part of which series.

Thinking about it, it probably makes the most sense to go with Monster World 3/Wonder Boy in Monster World as the third in the more ARPG/MV oriented part of the series, and WB3: Monster Lair as the second arcade action-style sequel (so a sequel to WB1).

MW3 is also quite good, though I do prefer MW2/WB3: DT in some ways.