Challenge Seeker

"There is no greater bore than perfection."

—Zaroff, The Most Dangerous Game

Some talented characters like to use their skills to drift through life as effortlessly as possible. Well, This Is Not That Trope, but the exact opposite.

Usually driven and ambitious, this character aims to constantly hone his skills by tackling challenges that push his abilities to the limit, and enjoys every minute of it. The harder the challenge, the better; they may even become bored with easy tasks or disappointed if something turns out to be easier than they expected.

Often a Determinator, because the more they get beat down, the more excited and fired up they get. May overlap with Hot-Blooded. Likely to Jump At The Call. Such characters often come across The Ace and find out there's Always Someone Better, but rather than create a bitter rivalry, they tend to be much more friendly about it, which may lead to Foe Yay.

Sister Trope of Spirited Competitor, but this type of character doesn't necessarily need a Worthy Opponent to compete against. It's more about the game/sport than who they play against, although they will usually be happy to have a worthy friendly rival who can challenge them and test their skills. Shonen anime protagonists in particular tend to exhibit this trope, especially those who "wanna be the very best."

Compare the Blood Knight, for those who specifically like a good fight.

Truth in Television - the Challenge Gamer is this in Real Life applied to video games.

Examples of Challenge Seeker include:

Anime and Manga

Endou in Inazuma Eleven. On more than one occasion, he actually got excited from seeing the opposing team score a goal on him or otherwise getting his butt kicked, because it meant he had a chance to play against amazing opponents. Hiroto also shows bits of this mixed with Spirited Competitor when he seeks out Raimon (and Endou in particular) for a match.
This is a lot of the stronger characters in the Pokémon series. Cynthia in particular is now at the point where she automatically identifies determined-looking ten-year-olds as godly powerful, which for her is reason enough to single them out for a challenge.
Kenpachi from Bleach, of the Axe Crazy Blood Knight variety.
One Piece has Roronoa Zoro, who wants to be the world's greatest swordsman, basically duels everyone holding a sword to prove that he's the best and improve himself.
From The World God Only Knows, main character Katsuragi Keima. He plays multiple games at once to mostly save time, but partly to challenge himself.
One of the defining features of Goku in Dragonbal. He even rejects sensible solutions for tackling enemies (e.g. shutting down the androids) in favor of taking on strong opponents face-to-face.

Comics

In Marvel Comics the Champion of the Universe goes around to various worlds looking for a good boxing match. His first defeat ever comes from The Thing, who refuses to stay down. He can beat the Thing but he can't defeat him, so the Champion concedes.
Also from Marvel Comics, the Grandmaster.

The Grandmaster has spent his extraordinary long life in pursuit of amusement through the playing of games. He has traveled throughout the known universe studying many civilizations' forms of games and play to the point of mastering them. He then began to devise his own types of tournaments and contests, challenging various opponents to games of skill and chance for high stakes.

Fan Works

Darius Walker from Honorable Hogwarts has shown a strong tendency toward this. He's fine using his wand against lesser opponents (or just poisoning them and sparing himself the tedium of a quick and easy fight), but when he heard what a good swordsman Jason was he came after him with a blade instead, to test his skills against the best.

Literature

Zaroff in The Most Dangerous Game, who says this is the reason why he started Hunting the Most Dangerous Game.
Valmont in Dangerous Liaisons does this as a Casanova.

Live Action TV

Soap: Jessica's attorney E. Ronald Mallu takes her case (she's on trial for Peter Campbell's murder) because it's so difficult. If he loses, it won't harm his reputation because nobody expects him to win, and if he wins it'll be a feather in his cap. Another attorney had said that he couldn't get Jessica acquitted with a mute prosecutor, her husband as the judge, and her family as the jury.

Video Games

Despite having ulterior goals, Augustus from Last Scenario treats his striving for power like a game.

I proved my superiority. I have already won. I Regret Nothing.

Vyse fits certainly the bill. In Skies of Arcadia, he repeatedly tackles challenges others deem impossible: such as escaping the Grand Fortress and crossing the Dark Rift. He even has a special ability called "challenge lover" in Valkyria Chronicles. Vyse is not only made of this Trope, he eats it for breakfast.

Vyse (to Marco): "Impossible" is a word people use as an excuse when they want to give up. Whenever someone says something's impossible, it makes me want to prove them wrong.

Othello, from Unreal Tournament III. From his backstory:

Othello was born into a family of means, and never had to work to support himself. Instead, he challenged himself with the most deadly jobs and in adrenaline-fueling adventures around the galaxy - climbing the ice floes of Mt. Karanish, participation in the Liandri Grand Tournament, even mercenary work in against-all-odds military conflicts.

Western Animation

The Dial M for Monkey episode "Rasslor". The title character comes to Earth and challenges its superheroes to wrestle him. He says, "For eons I have scoured the cosmos searching for the one adversary who could provide me with suitable sport." If none of the heroes can defeat him, he will destroy the planet. All of the superheroes go down to defeat except Monkey, who refuses to give up even after Rasslor beats him up.

Rasslor: I could crush your body, I could smash your bones, but I could never break your spirit. You are a marvel, little monkey! Any world that could spawn one as noble as you is truly blessed. I spare your Earth!

Stinkoman from Homestar Runner. "Are you asking for a CHALLEEEEEEEEEEENNNNNNNNNNNGGGGGGGGGEEEEEEEEEEEEE?!!!"

Real Life

Formula One driver Michael Schumacher, who believes "not being easy makes it [Formula 1 racing] fun."