One activity that I enjoyed a lot as I leveled up was to try and balance your threat on the very edge between your hunter and your pet, so that the mob is constantly shifting between the two. Ideally, the mob will spend a lot of time running back and forth without actually hitting either you or your pet very much. When you try this technique, your goal is mainly to take as little damage as possible, but really there are all kinds of ways to mix it up when you're leveling, and give yourself a challenge. You can try to achieve maximum efficiency of various sorts: fastest damage dealt, most efficient mana usage for grinding, even best ways to beat an elite enemy that should normally take 3 or 4 people to defeat. In this way, being a hunter feels like dancing or juggling when you do it well, and it helps make being a hunter more exciting and rewarding. (Of course, you'll need a threat meter to make this work.)
As a hunter, you also have many opportunities to use the environment to your advantage, more than any other class I've played so far. If you're the kind of person that likes to try out new ways of doing things, you quickly learn about strafing to one side or another, kiting enemies in circles, placing traps to slow them at just the right time, jumping off a ledge in order to force the enemy to run around the long way while you keep shooting at them.
All this you may try out with much excitement only to find yourself accidentally running into some extra enemies and having to fight several of them at once. Although technically it's bad form to attract more enemies in this way, it's not really the end of the world for you, since you have so many different tools at your disposal to deal with them all. In fact, it can really add to the excitement and challenge of playing a hunter by keeping things unpredictable and interesting, while at the same time usually staying within your ability to control the situation. (This is in contrast to many other classes, which can quickly spiral into disaster if you make the slightest mistake.)
In a group, also, there can be lots of interesting things to do. You make your pet stop growling in order to let the tank do his or her job, of course, but you still have to consider how much threat is going around all the time. You can sometimes go ahead and go crazy with your damage, then time your feign deaths, misdirections and other abilities just perfectly so that you never actually pull the monsters on to you directly.
You can also pull your pet back and forth at various times depending on which enemy or boss you're fighting and what kind of dangers it presents. On a fight like Pandemonius in the Mana Tombs, for example, there are certain phases in which you have to stop all melee attacks on the boss or else it will reflect a lot of damage back to you. It can be fun to try and time your pet's attacks exactly right, doing as much damage as possible in between these phases while taking the least damage (and keeping your pet alive too).
Also, when it comes to laying traps in a group, it can feel like a real work of art to manage chain trapping effectively. Brian already covered how traps work, but there's a special feeling you get when you do it just right, and you feel as though you can go on indefinitely. And if you can keep your damage up at the same time, that's even better. In this situation a hunter has at least 3 things to keep track of: the pet, his damage, and his trapped enemy. Juggling all that means that your actual dps may go down a little bit, but in the end, it's worth it because you're really challenging yourself to do a good job for your group. It's definitely not as simple as casting polymorph or shackle on an enemy again once the timer starts to run down, and the added difficulty makes it all the more rewarding to do well.
In short, although hunters have a bad reputation for being the "easiest class" to play, in fact there is a healthy range for a player to do either a mediocre job by just repeating the same old attack patterns all the time or an excellent job by applying a wide variety of techniques to varying situations - sometimes working multiple strategies all at the same time. That's what makes being a hunter special for me and keeps me coming back to this class no matter what other characters I play.