Since my new Alliance priest hit level 60, I'm feeling a bit lost. I do enjoy going on 5-man dungeon crawls with him, or to go raiding with my Horde priest. But these activities require that I have time for several hours in a row, and that the people I group / raid with are online. If I just have one hour, at some odd time, I'm not really happy with my level 60 characters. I hate soloing them.
The least bad soloing activity is gathering herbs with my level 60 warrior, or doing similar farming activities designed to earn me the money and potions I need for raiding. But even there I'm only willing to do it long enough to get my necessities financed. I still don't have an epic mount on any of my characters, because farming that much gold is too boring.
Another typical level 60 soloing activity is grinding faction somewhere. I'm revered with Argent Dawn with both of my Horde level 60s, but that is more due to them having visited Scholomance and Stratholme so often than to reputation grinding. My warrior made it to friendly with the Timbermaw, to get the transmute earth to air alchemy recipe, but then gave up on it. The others just farmed enough Timbermaw reputation to pass the tunnel without being attacked. My Horde priest made some effort to farm Cenarion Circle reputation in Silithus, but gave up soloing it when I found out that a raid clearing out AQ20 gave me 2000 reputation, while soloing 2000 reputation points would take me days.
My general problem with soloing at 60 is that I don't get the feeling that I am advancing my characters with it. Quests I can solo don't give items that are better than what I have. Farming gold to buy something epic from the auction house, or farming reputation to get some new recipe or reward, is an extremely slow process giving you very little reward for a very long grind. This is not a matter of wanting "free epics", it is a matter of having a feeling of character advancement and development versus a feeling of standing still.
So to get back into advancing, I'm leveling up another character. I didn't want to start from level 1 again, not before the Burning Crusade introduces new newbie zones. But my wife had a level 33 tauren druid on her account, her first character, which she hadn't played for a long time, and had no intention to play again. So as I'm paying for her WoW account with the same credit card as for mine, I was able to use the paid character transfer service from Blizzard to get the druid transfered to my account. So now whenever I have shorter play sessions and no guild groups lined up, I slowly level up that druid. I already did some quests, and I did like the ability to sneak past enemies as a prowling cat, getting directly to my quest target at the end of the cave, without having to kill all the mobs on the way. The obvious disadvantage is missing out on the xp for killing those mobs, so I will have to see how it works out in the long term.