Nowhere in the rules I have seen a statement about not being able to choose several prestige classes at the same time. And since there is no XP penalty when multiclassing prestige, it seems quite outpowered.
Assuming I meet all the prerequisites for several classes and have no difficulties finding a master, I could take 1st level of 2 or 3 of them and benefit from all the class base powers... Or am I missing something?
...That is, if the character qualifies for the prestige classes. A character who qualifies for multiple prestige classes can take levels in multiple prestige classes. In fact, it's often a good idea to do exactly that so you end up playing the kind of character you want. The DM can limit which prestige classes are available in his campaign, however, so asking first is always a good idea.
Characters get things from prestige classes, but as some base classes get almost nothing for staying in their base classes, that's not a big deal. Further, meeting requirements can be challenging for the more intense prestige classes, sucking up a lot of resources that can be put toward things a character might rather do or be good at, and classes that gain high-level class features (e.g. druid) will miss those when taking a prestige class. A prestige classes is a trade-off--often a very good trade-off but a trade-off nonetheless.
O, and just to be clear: I'm not saying there aren't crazy powerful prestige classes... because there totally are crazy powerful prestige classes. Instead, I'm saying the idea of taking multiple prestige classes isn't, in itself, unbalanced.
Some characters with a lot of prestige classes are overpowered. Some single-class characters are overpowered, too. Having more or fewer prestige classes says nothing about your overall power level, it’s all about which base and prestige classes.
So you can’t say something like, “oh, he’s got five prestige classes; he must be overpowered,” because that’s simply not true. You also can’t say, “single-classed? Must not be overpowered then!”
In fact, the prerequisites of prestige classes usually means its best to avoid taking too many, because prestige classes very often “tax” you for entry. Spending all your feats on weak things just to get into a bunch of prestige classes rarely works out in your favor. So when you say, “he’s got five prestige classes!” I’m thinking, “that was probably a mistake...”
But there are, of course, exceptions where it’s overpowered. But that’s very much the exception, and in such cases a character with just one of the prestige classes is still probably overpowered, just not as overpowered. Cases where individual prestige classes would be fine, but in combination there’s a problem are quite rare. It also happens, but literally only in a handful of cases, out of the stupidly-huge number of possible builds you can make in 3.x.
This can, by the way, be extended to multiclassing in general. Having levels in ten different classes doesn’t make you overpowered; it actually usually means you’re too scattered to be very much good at much of anything. And that’s assuming sane ways of calculating a multiclass character’s BAB and saves (i.e. Unearthed Arcana’s fractional BAB/save rules).
Also, for reference, very few people play with the multiclass XP penalty rules. They fail to accomplish what they ostensibly are supposed to do (discourage multiclassing), and their ostensible goal is widely seen as a bad idea in the first place (the breadth of material and relative ease of mixing and matching are two of the biggest strengths of 3.x). Also, groups with members at different XP values are a huge headache for DMs. So I wouldn’t factor that in much when comparing base classes and prestige classes. I cannot more strongly recommend against them; consider backporting Pathfinder’s Favored Class rules if it bothers you that without XP penalties Favored Classes are meaningless.
Nothing prevents a character from taking multiple prestige classes, as long as the character can meet all the prerequisites for all of them. Depending on the combination of abilities gained, it is even possible to use one prestige class to meet the prerequisites for another. The 3e version of Dragonlance even has a trio of prestige classes that depends on this: characters who rise through the ranks of the Knights of Solamnia are inducted into three knightly orders in turn, each of which has its own prestige class, and you'd be expected to advance partway through one class before moving on to the next. You can go back later, but you can't get all the way through all three without going well into epic levels.
However, as @HeyICanChan points out, this is not as overpowered as it first seems. One of the things the D&D designers focused on with 3.5 was resolving the level-dipping problem, and while the job they did was not perfect, most dips are not nearly as powerful as they once were. The strongest abilities are locked away until higher levels, and so although "omni-classing" will indeed net you a whole nest of powerful low-level abilities, they will all be low-level abilities. Specialized characters will simply get better stuff.
Now, it's true that extremely clever characters can still figure out ways to beat characters who outclass them power-wise. There are entire subgenres of literature dedicated to the idea. But there are limits, especially as the power difference gets greater, and in games like D&D, those limits are not nearly as high as they are in the books. It is unlikely that you need to worry about breaking the game.
It is completely possible if not encouraged
The primary barrier to entry for prestige classes are the requirements. Thus if you have the (usually) 6+ ranks in certain skills or have advanced through a class enough to have the required class abilities, BAB, or Saves, you have earned the right to enter the prestige class upon reaching your next level. So if the prestige classes are that different from each other, it will take that much longer before it's available.
Disclaimer
There are situations however where a prestige class might demand in-character training or reasons to join it, in which case your DM and especially the character's story development have significant sway, but ultimately this has nothing to do with the mechanics themselves.
