I've never really liked competing with others. As a child I would often refuse to play party games even at my own birthday party - I preferred to just sit out and watch instead. I suspect that this is because I actually have a fiercely competitive nature, and don't like the feelings that this nature inspires in me (the feeling that I must be better than others, regardless of their feelings, lest I be marked the "loser".)
For the same reason I've never really liked sports (playing or watching) and I suspect I would do better at work if on being told that I will be ranked against my peers my natural inclination was not to think "well I just won't play that game then - I'll just sit it out and do my own thing".
During one electronics class at school, I was helping one of the other students understand something that he was having trouble with. Another student advised me that I should not help him because we were to be graded on a curve, and helping one student implicitly hurts all the others. I don't want to live in a world where nobody helps anybody else - life isn't a zero sum game.
I think competition is overrated as a motivator for human accomplishment anyway. The great works of art of the world weren't created to prove their creators superior to all the other artists, and I think most of the accomplishments in academia happen in spite of the great competition for funding (and publish or perish rather than because of it.
I also suspect that the software industry could have achieved much more were it not for the duplication of effort caused by having competing companies solving essentially the same problems (particularly because of the exponential increase in complexity caused by having to interoperate). Different ideas should be allowed to compete on their own merits rather than on the merits of the companies that sell them.
Having a free market with competition to provide the best prices/best customer service/least environmental harm seems like a good idea in theory but from the individual customer's point of view, their only power is to take their business elsewhere. So my choice is between the supermarket that's close, the one that treats their employees well, the one that's cheap or the one that has the chocolate muffins I like. And (other than writing a letter that is likely to be ignored) I don't really have a way to tell the other two supermarkets why I'm not choosing them. The system only works on the aggregate level - individual consumers with requirements different from the profitable herd are basically screwed.
What's the answer? I'm not sure. Clearly some forms of competition are necessary (since communism didn't work out so well) and some people do great things precisely because of competitive pressures. But I think I would like to see a shift in policy towards rewarding cooperation and "absolute performance" and away from rewarding "performance relative to competition". Unfortunately that's rather more difficult to set up than a free market - in some disciplines (like computer programming) absolute performance is extremely difficult to measure absolutely (almost any metric you choose can be gamed). Also, if different factors become important (for example if we as a species suddenly decide than environmentalism is important) we all have to agree to change the metrics to take this into account, whereas in a free market we just have to have enough consumers decide "environmentalism is important to me - I will choose the environmentally friendly product even though it is more expensive".