Luigi's Mansion and Sonic Adventure 2 Battle.

Ikaruga. (best Shmup of its era and console)
Skies of Arcadia Legends. (best RPG on the console)

Resident Evil 1 Remake. I didn't play many games on the GameCube, but this game was phenomenal and I don't care that it's "just a remake".

If remakes don't count, then Resident Evil 4.

Two of the most defining games of my childhood/life in general. 

Luigi's Mansion is a bit too short, and Sonic Adventure 2 could've used a bit more polish in its 3 gameplay styles. 

Pikmin 2 narrowly beats out TTYD.

I thought in both. But Ikaruga is too difficult for me to recommend it. I remember the game lets you put your TV in vertical position, to play it like the arcade. In a small Tube TV from the era, that was relatively easy to do.
Skies of Arcadia Legends is a great RPG. I loved it. With some new content compared to the original... but losing the Pinta's Quest VMU game in the port, which makes Pinta character basically useless in this version. Oh, well.
Both games appeared before for Dreamcast (Ikaruga was originally an arcade Sega Naomi game, in fact).

I also always missed a Rez version for Gamecube, an amazing game shooter. PS2 got that port from the DC (GC instead got Skies of Arcadia, cancelled for PS2).

Technically... it's a remake. But it is the most impressive and astounding remake I and many ever seen, just 6 years after the original. it was so impressive, is almost like a new game. Those graphics were crazy in 2002.

Being Metroid Prime (I also recommend Metroid Prime II) and Smash Bros. excluded, as, I suppose, Twilight Princess also is, and knowing GC missed all the 3D GTA trilogy (and True Crime series are long forgotten)... I will choose 3 real exclusive games to recommend (unfortunantely, Gamecube has not the big names of exclusive games N64 had, and many of its games started to be just PS2 ports very soon, and some of its originally exclusive games were ported to PS2 anyways):

-F-Zero GX: This game, in 2003, was another exemple of the graphic crazyness GC could do, It was just incredible to see, at 60 fps. Made by Amusement Vision from Sega, the group ended to be the actual Ryu Ga Gotoku Studio. Amusement Vision was born as AM11, being this a split of the Sega's legendary AM2 studio.

-Star Wars Rogue Squadron II: Basically the same graphic crazyness, but this time, as a launch title! From the missed Factor 5, a great german company).

-Phantasy Star Online Episode III: The exclusive episode for Gamecube, this time based on a tactic board battles against monsters, using cards as inputs. It is some kind of addicting (I preffer its system a thousand times over Baten Kaitos Origins battle system... also using cards, but barely understanding anything there). The game has 2 well developed story arcs, from which you choose 1 at the start. Episode III is basically the ending of the original PSO story. It also had online gaming in Gamecube, thanks (again) to Sega, but this time the core gameplay was much more focussed on a single player story. Being online, you could interact and talk in the lobby with players of the Episodes I&II (released in Gamecube and Xbox), not only Episode III.

(PS: Phantasy Star Online also got an Episode IV (aka, Blue Burst) only for PC. It plays as the first 2 Episodes, and it is the direct continuation of Episode I & II, but not III. Episode III, instead, happens 20 years after everything, and uses a card combat system in a board style tactic battles, each battle placed on different locations).  

You also have exclusives as Eternal Darkness: I beat this game at least 6 times (you have to beat 3 times to see the extra ending), but i would prefered to see that game as a late N64 game, as it was planned and almost completed 2 years before). Almost the same story happened with Resident Evil 0, this last said to be only around 20% complete when N64 version was cancelled. You also have Star Fox Dinosaur Planet (the last game from the big Rare era, which ALSO was cancelled when N64 version was already completed and Star Fox had no relation in that. Fortunatelly, it was leaked some years ago). Star Fox Assault from Namco. And Pikmin 1 and 2 (but I never played them, i don't care about them).

RE4 will top any list on any console it have released on.

The Cube is home to many of my favorite games. Double Dash stand at the top of Mario Kart games. Melee as the most significant Smash game, Twilight Princess is all-around great.

But one game that needs the mention more than others is Eternal Darkness, since their are no other games like it and it has never been rereleased. It might not be as effective on modern TVs but the experience is so unique. Controls, graphics and camera have not aged fantastically but it is still i great game today.

There is a reason I get excited for every NAOMI port on modern systems. Legends is not the best version of SoA but still the best RPG on Gamecube. Modern ports of Ikaruga have more difficulty settings from free play to an easy mode. (It's still tough) but at least you can keep going forever. Only on PS4 and Switch.

Chibi-Robo
I love 3D platformers and Chibi - Robo is the only one that focuses as much on story as it does on exploration.
It is full of absurdness and campiness and, at the same time, tells a touching story of a problematic household. The dad is unemployed and watches super Sentai all day long, the mom is on the verge of divorcing him and they fight constantly, and their little daughter, affected by this situation, refuses to speak and only croaks like a frog.
The graphics are great but what really shines is the sound design. Every movement your little robot makes produces a different sound. The music is phenomenal and "Teriyaki Blues" is a treat.
Pity the game isn't well-known and also very expensive.

Not surprised about those new modes in recent Ikaruga ports. Game is too hard for casual gaming.

SoA Legends is probably the best "classic" JRPG game for Gamacube, yes. In animations like using stairs or open doors it feels a little mechanic and outdated, and graphics are nice and manga-based but far from what a Gamecube can really do, because the game was a faithful port of a 2 year old Dreamcast game, with some new content but not technically "enhanced". But the overall experience is fantastic. I always said that game is "the FFVII" for Gamecube. A beautiful adventure full of many hours of gameplay... and a little less darker game than FFVII. I bought my copy in 2002 or 2003.