I don't know if the kind of programmable IC you're looking for exists.
You can make your own simple audio player using micro-controllers like PIC, but you'll need to write the whole program, not just the song you wanted to play.

Depends on your audio quality, a very simple song (i.e. bepper type, sound quality is like dinosour-age nokia 3310 ringtone) can be produce with just a programmable micro-controller, the smallest and cheapest one is probably around RM10 (check lelong.com.my), not much component require. But the chip is quite big la, you can't really get those small circuit like they sell in bookstore. If you want better quality, like MP3 (i.e. instrument with you singing the song)....  That one more expensive, you need to get a MP3 encoder/decoder chip. Even I haven't play with that before. You can try google DIY MP3 circuit. This idea is best if you want to make a customize music box, cause the circuit+battery itself is just no sense to put onto a card...   

Oh I didn't know you just want some beeping music. I'm thinking using PIC to generate PWMs at the frequency of music "note" will do the job. Then you just need some driver circuit for the speaker.

Any pin count will do, you don't need all the pins. But need to check the internal memory. I haven't write a music program file before, my guess is even the smallest size will do.  Like tgrrr said, just engage the PWM, then change the duty factor and you can get your music already. See if I'm free to do a simple demo or not this weekend.  First need to get a beeper/speaker, maybe steal from my PC casing.