RAM with a SPI interface.
I've been wondering if such a memory device exists.

I'm intimately familiar with SPI Flash memory, but what about SPI RAM?

I've had a few project ideas for my Arduino that were hampered by the
small amount of internal RAM. Adding conventional RAM would eat up
all the IO pins and using flash or eeprom could potentially destroy the memory if it is written too many times.

True external RAM is supported on some AVR devices, and can be used like regular old memory. For those that don't (e.g., the ATmega168 in an Arduino), you might consider looking for what's available instead, but know that it will have to be independently accessed, one tiny part at a time, when you need it.

Avoiding EEPROM and flash EEPROM, you probably want simple static ram, or SRAM. I searched at Digi-Key for SRAM, which gave the choice of parallel or I2C interface. Atmel's version of I2C is called TWI (two-wire interface), and is available on the ATmega168, so you could use that. The part number that you're looking for is PCF8570, a 2 k static ram, about $2, and available in DIP or surface mount packages.