What would be an estimate of the value gained if the whole world started using Bitcoins? I remember that before the Euro was first introduced, I read estimates on how much economical benefit every person would get (GDP per capita) once the Euro was fully implemented. Estimates like these were used in the cost/benefit analysis used for deciding whether or not it was worth the trouble of creating the Euro at all.

I guess they didn't include the risk of a Euro-collapse and the related costs, but nevertheless, it was an estimate. I do not know whether this question is too speculative, but it is truly the most interesting question I can think of. If it would be too difficult to estimate for the whole world, then an estimate for a region would also be interesting.
According to xkcd: Money Chart estimated total economic production of the human race as of 2011 is $2,396,950,000,000,000 . Divide it by 21 million and the theoretical maximum price per bitcoin would be $114,140,476, but that figure is impossible, because for one, even in best case scenario Bitcoin will co-exist and compete with other currencies and would have only a small share of the pie. Second, Bitcoin is more efficient medium of exchange then USD, in other words, to complete transaction a someone required to hold bitcoins for shorter period of time therefore less total amount of bitcoins would be required for circulation and that also means lower price.