If you still mourn for the PSP, hackers might have found a way to get PSP games to run on the PS4. As an added bonus, those PSP games could even run with upscaled graphics. Though of course doing so wouldn't be "allowed."
Hackers found a PSP emulator hidden inside the PS4 re-release of PaRappa the Rapper Remastered. That's right, the PaRappa remastered was an upscaled PSP game, and user KiiWii posted their discovery of the PSP emulator on the GBATemp forum.
Turns out the PS4 remaster version of PaRappa ran on a PSP emulator but used a 4K texture package as a visual update. Using this emulator, KiiWii and other hackers were able to get several PSP games to boot on the PS4 by replacing the resource files in PaRappa.
Images showing PSP games like Loco Roco, Patapon, Namco Museum, and the PSP dev kit test app running on the PS4 were posted. However, the "PSPD" emulator isn't complete yet as there are still a ton of PSP games that don't work yet. Potentially because, as KiiWii says, there might be a 880 MB file-size limit on the emulator.
The PS4 is one of the more popular consoles to hack into, for a variety of reasons. Hackers have already gotten PS2 games to run on the system thanks to a loophole in the PS1 classics library, and now this. These are thanks to the emulators PS4 developers build for the console, which makes hacking it easier than on a, say, Nintendo Switch. Who knows what else the PS4 will be able to do in a couple of years.