To re-inforce N/a's post above, I've had as many as 5 routers set up with different bits at once and that wasn't for anything *big*.. just a coffee table.
A lot of it depends upon what you want to make... but there's a lot to be said for leaving a router with its bit set up as you proceed with the project, where feasible.
That way if you need to make another (or make a mistake on another part of the piece and need to remake it) you don't have to hassle with trying to match the setup.
Don't ask me how I know about the last part...
This doesn't mean that I'm recommending going out a buying a dozen routers at once... pick them up over time.. buying tools.. it's a slippery slope.
I'm a hobbyist that currently has 7 routers.. plus three on my wish list. For now I'm telling myself I'll probably max at around a dozen... For now. <g>
On the upper end, I know of a couple folks here in the
range... and others that make magnificant projects with one.
Truth be told, be sure you don't spend too much of your budget on routers and not have the $$$ for the wood and bits. Over time you'll likely end up with more in bits than routers.