Well, it's Star Trek Online.
This is a universe where people don't wear armor while running around helping farmers by dispatching monsters; a number-less character makes a whole lot of sense in this sort of setting. I'm not going to approach the technical issues, because I just don't have the space (or the knowledge) to try and write any of that out. This is purely a rough idea of how to do a massively game without numbers attached directly to characters. Or in other words, this is entirely theoretical.
Not everyone is going to like this approach -- namely the number crunchers -- but then this game isn't for them, is it?
Breaking down numerical goals
My STO would focus on Starfleet, because there's no point in trying to make things even more complicated than they're probably going to get. In the world (or universe, if you like) of Star Trek, it's all about ranks. Now most people would argue that ranks are a different way of having levels, but I would argue that it's all about how you give players ranks and what gaining a rank means. There would be no experience bar in Star Trek Online, instead players would be judged on their performance by Starfleet; which is also where they will get missions. This is where some kind of internal system would come into place, which would follow your performance in-game differently from how it's normally tracked in current MMOs.
For instance, normally in a massively game players are given a goal like, "Get 100 points of experience to reach level two." At this point they need something to do to get those experience points. With just about every massively game out there, that means fighting creatures, defeating them and upon defeating them gaining five or six experience points.
What I propose is that STO would remove the primary goal of simply defeating an enemy. Instead, the primary goal would be to work together as a team to meet mission objectives -- whether they are surveying a new solar system or fending off a Borg attack. The overall goal would be to develop a system that would keep track of players' achievements without tacking numbers into a window or a bar.
Say you're an engineer. It's your job to keep the ship running at maximum efficiency. So throughout any given mission the game is tracking how well you handle your job by observing how quickly and effectively you respond to your particular ship's needs. At the end of every mission, the game would take a snapshot of how well you did overall. Looking at things like hull integrity and how rapidly damage was repaired -- the game would judge performance on several different areas. It would monitor power-core stability and how fast you managed power diversion when asked for it by the captain. After a given mission, you would receive some kind of reward based on the difficulty of the mission.
Mission Ranks and how they figure into the non-equation
Every mission has ranks: S -- A -- B -- C -- D
These ranks determine many things about a mission, but their main functionality is to help developers and players judge the challenge and rewards included with a mission in an easy way. So for instance, completing a Rank S mission will generally guarantee a promotion within Starfleet, but sometimes will reward players with medals of honor or distinguish -- you'll never know exactly which it will be since it's partially based on your performance.
Anyone can join any ranked mission no matter their actual standing in Starfleet, be it lowly ensign or experienced commander. However -- and this is breaking lore a bit, I think -- the captain has final say as to who can or can't join the crew for any given mission and crew members are interchangeable mission-to-mission. Of course failing a mission only costs your crew time, unless a more interesting (but not demoralizing) method could be thought up this seems like the best solution for failure. Since Starfleet owns the ships themselves, it's not like players can have repair costs for their ships.
Keeping the balance: Starships can have numbers
Which brings me to the perfect way to keep combat balance in an MMO without numerical value attached to characters in any way. Ships can -- and pretty much have to -- have stats attached to them, yet ships are not something attached to any specific player. They offer tactical transparency to players, which is a very important part of balanced combat.
Sure, a vessel in Star Trek Online could be described as a piece of equipment, but it can be so much more than that. Ships can be personalized by guilds -- special insignia, ship names, interior coloring and design and outward appearance tweaks. It would be tempting to try and create massive ships and use them as housing for entire guilds, but that's a mistake.
Instead of thinking about ships as super-massive, they should be thought of as only fairly huge at best. It's hard enough making the engineering or science officer class interesting without adding five more to compete with on a massive ship. What would be far more interesting is large fleets of ships controlled by a guild. These ships can all be customized in appearance for a sense of cohesion, but not in statistical value. What makes the difference in battle are player skills and in what ways they're used.
But skill use is a whole other article for another week.
Guild housing should be something special and in a game like this, giant spaceships are going to be pretty common. So for guilds with fleets of ships, it only seems logical to give players space stations for guild housing. Obtaining one could be a part of completing a S-rank guild mission.
There are plenty of other pieces to consider, but the core of the non-numbers game is there. Although it may only be a pipe dream to hope a developer could pull something like this off -- it's still nice to dream.