How many mp3 songs can the PSP hold?

There's no exact way to say how many songs a psp can hold. It depends on how much memory you have, the format of the songs, as well as the length of the songs. Most people use mp3's, and the average song is only about 3 minutes so about 3 megabytes. These are the assumptions we'll use.

PSP 1000/2000/3000
Regular PSP's do not come with built in data, but you can use a memory pro stick duo OR you can use a combition of 2 microSD cards and microSD to Pro duo adapter (The combo may require you to purchase more items, but it actually is cheaper right now than buying a pro duo).

Pro duos can theoretically come at sizes of 2 Terabytes in the future. However for now I will only consider up to 32GB.
2GB Pro duo: Max of 666 songs
4GB Pro duo: max of 1333 songs
8GB Pro duo: Max of 2666 songs
16GB Pro duo: Max of 5333 songs
32GB Pro duo: Max of 10666 songs.

So the PSP 1000/2000/3000 can potentially hold 666-10,666 songs according to the size of the memory card you bought. However the more games, pictures, and videos you have, the less music you can put on.

PSP Go
The PSP Go comes with 16GB of flash memory. In addition you can buy Memory Stick Micro (M2), which can only go supposedly go up to 32GB (though the current max is 16GB). So while the maximum total memory for a PSP Go right now is 32GB (16 + 16), later on it could go up to 48 GB (16 + 32).

No memory Card: With only 16 GB, you can store 5333 average MP3 songs. This does not count games, videos, and save data you may also have on your PSP. The following cards also have the assumption that there is nothing else on your card.
2GB M2: You now get to have 666 more songs, so a total of about 6000 songs.
4GB M2: This gives you about 1333 more songs in addition to the built in memory, so a total of about 6666 songs.
8GB M2: With this card, you get 2666 more songs for a total of 8000 songs.
16GB M2: Oh my this can give you 5333 more songs for a total of 10,666 mp3's!
32 GB M2: While unreleased yet, this could give you a whooping 10,666 extra songs for a total of 16000 mp3 songs.

So the PSP Go can hold anywhere between 5333 and 16000 songs depending on how much storage you have. However if you have videos, games, and save data, this will cut into your available space!

Note: While the PSP Go may hold more music, keep in mind that a M2 card is more costly than a pro duo card. In addition, use the SD adapter combo is even cheaper than buy a pro duo. So if you want to buy a lot of memory, a regular PSP will be less costly.

For example: (let's say you want 16GB of space);
PSP Go + 0GB M2 = 250
PSP 1000 + 16 GB Pro duo = 130 + 65= 195
PSP 1000 + 8GB microSD + 8GB microSD + microsd to pro duo adapter = 130 + 20.5 + 20.5 + 8.3 = 179.3
PSP 2000/3000 + 16 GB Pro duo = 170 + 65= 235
PSP 2000/3000 + 8 GB microSD+ 8GB microSD + microsd to pro duo adapter = 170 + 20.5 + 20.5 + 8.3 = 219.3

Prices are the current prices as of October 15th 2009. The cheapest price + shipping from amazon.com or buy.com was picked. Retail prices (like best buy) for the memory cards would be even higher. Most items came with free shipping except for the adapter ,which was about 2$ + 6.3$ shipping hence the 8.3$ price tag

Sources:

Max size of Pro duo

Max size of M2

16 gb pro duo price

8b micro sd price

micro sd to pro duo converter price