Removing the level cap would wreak havoc on the balancing though, and is a terrible idea. It was seen clearly in TLA before they introduced a level cap, where "god characters" would roam towns killing everything on their own.
Yes it's terrible idea, that's why we gotta think of another way.
If you just kept playing a single dude, doing various stuff like quests, crafting, pve and pvp you would collect some sort of alternative exp or reputation that you could spend to special benefits. You had to do all of those, not only one to get the benefits. You could have weight on one of them more than others, but doing all of them at least a little would be necessary.
What would these benefits be? Temporary buffs to stats and skills, temporary perks, unique items depending on your profesion and unique companions.
Where would you get them? From npc factions.
What exactly was needed to do for them? Various stuff. Let's say you're ncr ranger. Rangers want you to kill slavers, you get ranger +rep by killing slaver players and slavers, freeing slaves, waging war against metzger, providing rangers with weapons, medicine and amors that you craft. You couldn't just grind slaver encounters because the +rep from doing it would diminish, so you'd have to craft some weapons or armor. Crafting weapons and armor doesn't help because the +rep production diminishes from all crafting, so you'd better just craft highest tier weapons, armor or medicine. Once you can't get much more rep from those two above, you should go hunt some slaver or raider players, but you can't hunt them forever because +rep from them would diminish too.
Now that you've got some +rep you can buy a temporary benefit from the rangers. You lose it when you die, so don't die, better team up with other rangers.
If you die in the hands of slavers or other enemies of rangers, you lose your rep and status in their eyes. And don't even try to ~cuttheveins, because the rep is lost when you lose your Ranger star badge, but you won't get any +rep when not wearing it. This badge will make others able to recognize you as Ranger.
This would keep lvl21 dudes busy forever, I guess.