This is actually quite a few days old, but I've been... busy. To say the least.
Surely you're all aware of the downloadable PS1 games being made available by Sony on their online Sony Store site: with the use of a PS3, you can download several PS1 titles and transfer them to your PSP at a couple of bucks a pop. The games are fairly faithful to their original versions due to the fact that they are in fact emulated on the Playstation Portable. However the selection of titles available is poor at best (mostly first generation games), but with the promise of more on the way.
Cue the PSP hacking community, who've been fighting for the right to run unsigned code on their PSPs for homebrew applications and user-created games. With every new firmware update, Sony has managed to tighten the security features of the PSP making it difficult for these hackers to exploit (we're up to v. 3.03, I believe).
For a while now, a programmer known as "Dark_Alex" has been releasing "custom" firmware that can be flashed to your PSP, giving it the ability to run unsigned code ala the old firmware 1.5 PSPs while giving it the features of newer firmware (including the ability to play PSP games requiring higher firmware versions to run). The flashing itself is tricky and requires downgrading your PSPs current firmware to 1.5 (as of right now, anything 2.80 and above can't be downgraded) and then flashing the custom firmware.
Recently, D_A had released "Firmware 3.02 OE-A", giving those who use the app the ability to run PS1 games purchased from the Sony store, just like the official 3.00 firmware did when it was released. However D_A promised a new feature for his next update, and just a few days ago released 3.02 OE-B, a hacked 3.02 that allows you to run any PS1 game via the official Sony emulator. That's right, any PS1 game can be loaded off your memory stick and run thanks to this bit of code.
It's not exactly plug and play, however. It requires certain files that can only be obtained from PS1 games purchased from the Sony store (the actual emulator files, I reckon), ripping them out using software D_A includes with his firmware hack, and applying them to an image file of a regular PS1 game. The newly created PBP file is then uploaded onto your memory stick and then run from the PSP boot menu. And for the most part, it works quite well. Most games run flawlessly or with some minor sound/graphical issues. Unfortunately there's no analog support at this time. The Analog nub is used to simulate L2, R2 or both, and can be adjusted to act as the D-pad if you wish.
However, it is near impossible to test the entire PS1 library in such a short time so emulator's compatability is in question.
What Dark_Alex has done here is nothing short of astonishing. In one day the PSP's potential library has grown by about 400. However, one has to acknowledge the dubious legality, not to mention danger, of such an application. Firmware hacking/flashing is incredibly risky. Screw up and you effectively "brick" your PSP, making it worthless. And don't expect Sony to honor your warranty if you do.
As to the issue of legality, well... Sony surely won't be pleased to learn that their emulator has been compromised. Already the big file sharing and Bittorrent sites have PS1 games pre-formatted to play on PSPs using the D_A's OE-B firmware. But is there anything that they can do to stop it, short of releasing newer firmware (something that won't be an issue to those who've already switched to custom firmwares)?
Probably not. Burnable PS1 game images that can be played on emulators and modded systems have been floating about for years now, and PS2 games have since followed. Piracy is rampant on the net, with little chance of stopping it. Once the ball gets rolling, there's little anyone can do.
So, how does this affect you? Well, if you're sporting a brand new PSP... it doesn't! As mentioned above, there's currently no way to downgrade PSPs running on newer firmware (2.80+ and up). And those who've had their PSPs since launch but have upgraded their firmware at every opportunity are also out of luck. Nay, for the most part this news really only affects those with the sense not to upgrade everytime Sony shoved a new update in your face (whether through its Network Update or from new games like MGS: Portable Ops which carry upgrade files and even require them to run).
For the rest of us who were willing to risk a brick in order to have a PSP with no self-imposed limits... it's been one great thing after another, and this new customized PS1 emulator is probably the greatest thing to come down the pipes since the SNES and Genny emulators. Currently, I'm replaying Castlevania: SotN at work, and I've also tried out Klonoa, Final Fantasy 9, Um Jammer Lammy (weird timing issues, that one) and Strider 2.
So no, I guess I don't regret buying a PSP.
However the selection of titles available is poor at best
But it's Ridge Racer!
Be that as it may, it still stinks.
...
I can't help but think that I'm the only one excited about this.
...Or I would be, had I not upgraded my PSP to the point of no return... 
Portable Jumping Flash would make me a happy man.
I think it sounds pretty cool, although I don't own a PSP, and don't ever expect to. Portable Armored Core, Gran Turismo, and Mega Man X4 would be pretty nice though. (Armored core might be a bit tricky to play on the PSP though)
If I could get hold of a PSP with the right firmware, I'd might actually take it over a GP2X(Currently the High Priced Handheld I'm most interested in), as this totally blows GP2PSX(The GP2X's Playstation Emulator) out of the water.
I must say, I'm impressed with what the PSP Homebrewers have accomplished, despite Sony's efforts to put them out of business.
For those who are curious, there's an ongoing compatability thread here. As you can see, NTSC region games dominate the list.
As Zerosky surmisses, games that require Analog sticks for basic control (Armored Core, Ape Escape, etc.) will prove to be unplayable. Anything that can be played using only the digital pad should be fine. But the emulator is far from perfect, and crashes that could keep one from progressing may occur as games are further tested (Resident Evil 2, for example... it crashes when you arrive at the police station).
Also, there's the issue of multi-disc games. The emulator does not allow for swapping discs (images, in this case). However, many multi-disc games allow the player to save their progress at the swap point. These games should be playable all the way through. Examples include the 3 Final Fantasies and Xenogears. However other games don't do this, meaning at this point it is impossible to proceed past Disc 1 of Metal Gear Solid, Parasite Eve and others without the use of a PC emulator and a memory card converter that can handle the popstation save format (currently, there is none).
Gyser: It wasn't that long ago when no one thought 2.00 would be cracked. The hacking community is tenacious. If there's a way, it will be found.
On the subject of Armored Core, the PS1 versions never used the analog sticks actually. The reason I though it might be tricky is because all four shoulder buttons on the PS1 controller get used frequently. R1/L1 are used for strafing, and R2/L2 are used for looking up/down. I'm not sure how easy it would be to look up or down while moving if you have to use that analog nub to do it. Could be doable though.
Aha, I thought there was something odd about that. I haven't played the first Armored Core for years. Duly noted. And yes, that would be quite awkward.
