A great way to keep yourself entertained at max level is to start a collection. Most online games have any number of things you could collect for fun. This collection can be anything you like -- it doesn't have to be something that actually helps you out in-game (in fact, it's probably more fun if it's not). What better way to flex your newly earned levels than with a virtual egg hunt through lower zones for whatever interests you? I've known players who collect rare swords, pets in World of Warcraft, mounts, holiday items, or interesting vendor trash.
Your only barrier here is your imagination. You could go digging for items to make yourself unique costumes to wear in cities (this is even more fun in games like Lord of the Rings Online or EverQuest 2 where you can use an appearance tab to wear your costumes anywhere), search for the grossest random animal parts you can think of (diseased bear pelts and basilisk eyeballs for the win), or build a collection of trophies from fallen foes-- you probably don't need that Onyxia head quest reward anymore, so why not just drag her head around for fun? Collecting items is a fun activity you can participate in by yourself and then share your collection with your friends later on.
Searching for "Clickies"
This is an off-shoot of collecting (okay, so it's the same thing), but I include it as a separate category because I find it particularly fun. Most games have items that you can keep in your inventory and right click to produce special in-game effects. If you amass a small collection of these, you're sure to be the life of every party and the bane of every raid. Even if they're one-shot items that you can stock up on, it's a blast to have fun toys to play with.
While most of the clickies in WoW are strictly for fun, a number of EverQuest's clickies were extremely useful. I loved my Mage's Eye of Zomm bracelet that you could quest for. It let you summon a small eye that you could zoom around corners with and scout ahead in dungeons, with no worries about getting aggro. Other items of note include the Orb of Deception and dancing piccolo from WoW, all of the race-changing items from EverQuest, and throwable snowballs that appear in multiple games during the winter holidays. I'm sure there are plenty of examples from other games.
If you play on a game that has open PvP, there will always be people harassing lowbies. If you're bored one afternoon and want to feel like a hero, take your leet self to zones where you know would-be griefers and gankers are hanging out and go a-hunting on behalf of the slaughtered levelers. It helps a lot to offer your services in general chat and ask if anyone is having problems. Most of the time, people will welcome the assistance and hail you as a stalwart hero and defender of the weak!
It can be fun to serve as a guardian for lower players and help them walk NPCs through dangerous and frustrating escort quests, or turn the tables on the griefers and corpse-camp them for a change. Working the anti-grief beat is a great way to go find yourself some fun PvP, help your fellow players, and develop a good name for yourself on the server all at the same time. When the griefers go cry on the forums about how you're making their job hard for them, it's nice for a good laugh, too.
Even if you don't play on an RP server, never underestimate the power of a little creativity and a few friends that are willing to participate in a fun afternoon. There's absolutely nothing stopping you from treating your game client as a make-shift tabletop RPG simulator. Designate one friend as the "GM," and have him make up some interesting quests for your group. Using WoW as an example, you could all chip in 50g and have him buy a low-level epic item from the auction house that's appropriate for one of your alts (or just fund it yourself). Then, have him send you on an elaborate questline to "build" this "ancient weapon of great power" and vanquish some powerful foe.
Maybe this could consist of escorting a level 40 character into dangerous zones like BRD or Stratholme to acquire whatever items the bosses of those zones drop. You return with your prizes to your GM (disenchanted into the proper materials, of course), who uses them to "build" a Dazzling Longsword for this level 40. Then you take him, with his new weapon, to defeat a powerful named NPC in a dungeon appropriate for his level (keep him healed while he solos a boss, thanks to the "magic" of the sword). A lot of players will think this is corny and contrived, but it's just one example of what you could do with a creative group of people. Don't underestimate the power of your imagination when it comes to having fun!
Depending on the game you play, you might never have even considered picking up a tradeskill. "Crafting? How boring." But tradeskills can be a lot of fun and give you one more way to advance your character. Some games, like Vanguard and EverQuest 2, have made tradeskills into a mini-game in their own right. As the saying goes, "Don't knock it 'til you've tried it." You never know the fun you might be missing.
In addition to the pride of using items you make yourself, tradeskills are often a great way to make money in games. Do some research and pick a trade whose items are highly demanded, or a trade that produces consumable items. With a little effort, you can turn yourself into a little self-contained factory and corner the market of your choice. In fact, that leads directly into my next suggestion...
Playing the Market
The World of Warcraft maximum gold cap (just over 214k gold) was recently discovered by a German player. How did he do it? By playing the game like an economy simulator, of course! For some players, monitoring the market pricing fluctuations and taking on the task of buying low and selling high (or "flipping," as some people call it) can be an extremely interesting sub-game within the larger framework of your favorite game. While it's certainly easier in some games than in others (Auctioneer, anyone?), you can get a feel for the market prices of any game simply by watching and participating in trade. Over time, it's hard not to get a feel for it.
Once you've been bitten by this particular bug, it's hard to stop. Trade happens all the time, every day in online games, so you essentially have a new, fascinating minigame that's always available to you. After a few weeks, you'll be negotiating with the best of them. It gets really interesting when you can amass enough capital to begin experimenting with things like controlling entire markets for a good, or artificially inflating demand to insane levels by buying every single item of one type on the market and then doling them out sparingly. If you'd like to get started with this and you don't know how, I highly recommend Og's Ledger. While it's World of Warcraft specific, many of the tips can be applied to markets in general.
All of these suggestions are just a few of the ways you can keep yourself entertained at maximum level in an online game without falling back on more of the usual "grind, grind, grind." I'm sure that there are tons of other creative ideas out there and things that players like to do which fall outside of the intended play parameters, and I'd love to hear what you like to do to alleviate the boredom.
Next time you're sitting around at the level cap, bored as heck and with nothing to do, try giving one of these a shot. You never know when you might discover a whole new exciting aspect of a game you thought you had done to death.