I like the simplicity. No excuses, no mind games, no luck, just you and the weight. It means that even when a lot of other things are stressing you out, you can go to the gym and achieve something which you can't question - if you make a lift, that's it, it's done, and it can't be taken away from you. When you can see that you're lifting bigger week after week, doing things that you know you absolutely couldn't have done a month ago, it's a great confidence builder and teaches you the value of hard work and perseverance.
I've only competed properly in powerlifting once, but that was an amazing experience - the atmosphere really brings out the best in people.
I'm not a lifter - but I enjoy exercise/training purely for the fact of 'skill acquisition'.
Being able to take a skill, break it down, master it, and move on to the next skill; it motivates me a lot and it develops key personality traits that can't be learnt from an A-Level text book such as; delayed gratification.
I like being challenged; exercise plays it's part here in many ways. I want to keep exercising way past the age of 70 - also I do it partly to set an example to other people around me. I have no idea why some people hate exercise with a passion etc