Non-Volatile Serial Memory Chip
Probably not, but then writing the simple code for this in a microcontroller is really easy.  EEPROM chips don't work this way because there is more to a EEPROM than just storing and reading back a string of 500 bits.  Some SPI EEPROMs come close to what you want once you get going, but there are still issues of pages, opcodes, and the like.

500 bits is only 62.5 bytes, so any micro with at least 64 bytes of EEPROM built in can easily do this.  That is a small amount of EEPROM for a micro.  This also allows the button debouncing to be done in firmware.

Also, 500 bits by hand?  Seriously?  The chance of getting those all right is pretty much zero.