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For the most part, you will do fine with the core rule books only, but there are some char builds that are "lagging behind" without using splat-sources...  I'm of the opinion Paladin make lousy defenders (though still strong well-rounders) without Divine Sanction (included in Divine Power).  Wizards also are just... well... lousy.  At least in terms of at-wills, without their various splats.

If your players are crunch-savvy though, this shouldn't really be a problem.  Should be fairly obvious which classes/builds are lagging behind (Back when we only had PHB1, we figured out pretty quickly that Chaladins couldn't defend worth a damn, since their MBAs were bascially garaunteed to stink)

I've been running SCAP for about 2 years now.   We had got to Chapter 4 when 4E came out and I decided to convert to 4E.  Back then we obviously only had the PHB, DMG and MM.   The players all converted their characters into PHB characters of 5th or 6th level (can't quite recall) and off we went.

We are still using only PHB rules for the players, although I have started to give them equipment out of DDi (I have a subscription as you can see).   I keep abreast of all the latest DDi bits and I throw monsters and traps at them from any DDi publication I fancy.   But they stick to the PHB.   They all own a hardback copy of the book and are happy to explore this book before moving on to other supplements.  No one has died in an unrecoverable manner yet so there has been no need to make new characters at all.  I think if that happened I might open up PHB2 (which I own) but would probably prefer them to just pick a new class from the PHB which has not been tried yet.

Actually, I have just recently bought all three of the first Powers books and have said they can start using stuff from those now when they level up (and with retraining).  But so far there has been no interest particularly.

The campaign is currently sat at 16th level and I can honestly say it has had zero downside to restricting the PCs to using PHB only material.   Equally I don't think things have ever got unbalanced with the use of monsters and traps from all over 4E.   I am pleased to say that so far I am not seeing the power creep I did when the 3E splatbooks came out.

But from our perspective, sticking to the PHB for characters for this campaign means that when we start up our next campaign we have a whole raft of new stuff to play with which will make that campaign a completely fresh game - that has to be a good thing.

I really think you could run an entire campaign with PCs sticking to PHB1 for a year.  Then start a new campaign with PCs using PHB2 the next year.  And so on.   I reckon it would work fine.

Cheers
Blakey

'Core only' was a great idea in 3.x, which was overloaded with the possible combinations of multi- and prestige classes, feats, and so forth.  4e is, if anything, pumping out supplements even faster, but with the more restrictive multi-classing, and with many feats specific to race/class/build, the dis-synergy is more subdued.  Also, in 4e, 'core' is broader than it was, so you'll still have almost all classes available, for instance.

In general, I'd say that 4e works fine if you want to restrict it to certain books (or certain power sources, say) for a single campaign with one set of characters.  Over the long haul, though, each class is not really good for that many different PCs within the same group (it'd be pretty tough to make a 3rd greatweapon fighter who doesn't seem a lot like the previous two, but it's easy enough to make a Warden that's unique compared to previous non-warden defenders played in the same group).

As a DM, I value the AV and AV2 for treasure generation, so you should keep that in mind. The treasure in the PHB is very limiting in regards what kinds of items are available at each level.