If the bitcoin (BTC!) becomes the world reserve currency, then banks will long have adopted it. They will have scared people because of security issues when holding BTC by themselves and they will have made people deposit and transact via the banks, which in turn allows the banks to work with IOUs instead of actual BTC.
This will result in the ability to massively inflate said IOUs by just issuing more of them as debt which will have to be repaid (plus a little extra). For this to not get out of hand an institution will be appointed bitcoin reserve bank and it will be able to control the creation of said IOUs on a pace they like.
As a result, the rich will get richer and the poor will suffer.
The value of a bitcoin by then? A lot less than now because it has been massively inflated with trillions of bitcoin IOUs everyone is forced to accept as settlement of debt.
In that case, a bitcoin IOU won't be worth nearly as much as a real bitcoin. Do you remember gold backed US currency? It use to be a gold IOU.
Thats how it should be in a free market, but even today a $1 note is worth the same as a virtual $1.
If you are forced to accept IOUs as settlement of debt, then it's worth the same as the real asset it's supposed to represent as people can just not pay up in actual BTC and settle the debt later on with IOUs.
An increased value of actual BTC could be the case if people predict the system to fail soon and want to get something they can use even after that happened, but that would take lots of time and by them bitcoin would probably have been replaced.
Personally I value real gold a lot higher than gold IOUs (which I think aren't worth anything as it's well known there not enough actual gold to pay up on them) but most people think it's basically the same as they are told it's backed etc.
Fortunately we are seeing a shift in that mentality. It's slow but it's there. However, if we let bitcoin IOUs become the norm regarding bitcoin payment processing, we won't get out of the system. And it's not like thats far away. it's been done for quite some time by Gox and it's highly likely there are others doing the same as we speak.
Why wouldn't someone print free money if he has the option to do so? This is a scary scenario, I know. But it is what can happen and it is what people will accept as a given if marketed correctly and there is no counter pressure.
We have to be extremely careful in that regard these days. We as early bitcoiners owe our children a system that is a lot less corrupt and a lot more enjoyable than todays system. We have done a great job with the basics, now it's out job to not lose the free currency we all came to love. It has been done once with the Internet, the free speech channel and it can be done again.