The class system is the traditional way to define who and what your character is, but there are other methods, as proven by these RPGs.
One of the most freeing games to play is Dungeons & Dragons with a group of dedicated friends around a physical or digital table. Players can fully customize their character and make them into the hero or villain they want to be, complete with a class and personality. Class systems evolved from D&D, and while no console or PC game is as freeing as a good old tabletop session, there have at least been fun games with classes to play with.
Some great examples include Final Fantasy Tactics, Diablo 2, and Metaphor: ReFantazio. While classes often coincide with RPG systems, some games offer more dynamic ways for players to define their characters with skill trees or equipment without being set in one class. To get a better idea, these are some examples.
The Elder Scrolls 5: Skyrim
The Elder Scrolls 5: Skyrim lets players create their character, including several bizarre races like the cat-like Khajiit, but there is no class system, unlike the previous game, The Elder Scrolls 4: Oblivion. Instead, players can forge their own path and scale their character into a form that best suits their gameplay style.
Everything players do will level up, including just walking around, which is a traditional part of some entries in The Elder Scrolls series. If players want to be a mage, then they can learn spells, boost stats toward Intelligence, and gain skills in that respective tree. Using spells repeatedly will increase their potency, and this is only one example of how players have complete control over their character build.
Fallout: New Vegas
Fallout: New Vegas gives players some basic choices for character customization visuals, followed by stat allocation and a few Perks. When players level up, there will be certain tiers wherein players can increase stats or gain Perks, which is the highlight of the game. Both stats and Perks can scale a character toward a certain class archetype if they want to play a certain way.
For example, if players level up their Strength stat and gain Perks like Super Slam, Nerd Rage, and Piercing Strike, then they will do more damage with melee attacks. With the various Factions players can align with and the branching leveling paths players can take, Fallout: New Vegas has a lot of replay value compared to most Fallout games.