Having done a little research, I have come to the conclusion that many Giffgaffers, like myself, quite like the idea of competing against one another in a light-hearted sort of competition. The problem seems to be that people suggesting the ideas make them too specific, and consequently there is always someone to object to it. I shall attempt to condense the general idea of most/some of them here, and also summarise my thought on various ideas.
Adverts - People seem quite eager to make them, but want to get recognition for going to the effort of doing so. That seems fair enough. Obviously a winning advert would not be going on TV, because of the massive expense that would represent to the company, but I'm sure gg can think of something to do with a well made advert that a user has submitted.
Plausible Prize: Free goodybag?
Recruits - This is one idea that would be particularly easy to judge the winner of, but for the everyday Giffgaffer, such as myself, it seems immediately obvious that I would not win (so there would seem little point in trying) because I know that there are those on the network who own phone shops and send out SIMs with their phones, and consequently get a lot of new recruits. So while this idea may not be of interest to me, it may be worth looking at as a competition between those in the afore-mentioned advantageous positions. This idea is already down as "Accepted" in one form, here. The point has also already been made that gg already rewards us for bringing new recruits, so it's not exactly necessary to supply more motivation.
Plausible Prize: V.I.G. Status?
Other - In this I am including all suggestions like poetry competitions, and sunflower-growing competitions, and whatever other sick things you people get up to. These ideas don't contribute in any way to giffgaff, but they would be fun to do.
The Prizes Problem - Without the promise of a prize for the winner, people are less likely to take part in competitions and consequently the whole idea might just fall on it's face. However, there is nothing to stop people from starting up their own competitions with either no prize, or a small prize for the winner which the Quizmaster would have to supply. Perhaps this would work better if giffgaff were particularly generous with points for people who organised competitions, but why should it always have to come from giffgaff? If you want competitions, arrange competitions. Up to a point, it really is that simple.
It would be simpler still if giffgaff supplied us with a competitions forum board so that they were all in the same place, but I imagine it would have to be clear that it would be worth it and people would use it first.
However, assuming we can't rely on the Men and Women of Giffgaff to supply their own competitions and prizes, we could do with giffgaff setting something up. In the case of the competitions relating to recruits and adverts, it doesn't seem unreasonable to ask giffgaff to supply some sort of prize as it is in their benefit. In the case of all of the other arbitrary stuff people would want to compete on, there seems no logical reason why gg would want to reward us for being good at poetry. However if there was a small entry fee to such competitions, perhaps this idea could work. So in order to enter this imaginary poetry competition you had to agree to have 5 points taken away, or else have 5p deducted from your account, and this would go towards the prize for the winner. This would also mean that such competitions would not be spammed with people entering any old rubbish in the vague hope to be given kudos or something. The only problem remaining would then be that someone would have to sit down at giffgaff HQ and read all of the entries and decide (most likely with a reason) which one had won and why, then sort out postage for this elusive prize to the smug winner. All of this takes up someone's time. A much simpler way of judging would be to modify the kudos system on the (currently non-existent) competition forum so that it worked as a poll-type system, and the entry which received the most votes from the community would obviously win. This would make it much easier for other users to organise competitions too, as they would not be burdened with the responsibility of playing the objective judge.
In Conclusion - In light of all of this, it does not seem unreasonable to ask giffgaff to take a couple of steps towards helping us achieve this goal of having occasional competitions. The competitions forum I talked about earlier would be a good start, and a really nice way to start it off.