I remember that there were some PS2 games that required the Hard Drive to be installed (either PCMCIA or Network Adapter).
Which games required the Hard Drive to be work?
This list on Wikipedia seems relatively complete, but as it's quite an expansive list, I'll post some notable highlights here.
Final Fantasy XI appears to be the only NA game that truly required an expandable HDD in order to accommodate for various patches and upgrades. XI was an MMORPG, and while most MMORPGs at the time were on PC and had access to hard drives for patches and upgrades, the PS2 did not natively contain an HDD, so the expandable drive was necessary. XI was also available as a bundle with a 40GB HDD in some.
Other notable games included a myriad of ESPN sports titles that utilized the HDD to support and improve replay functionality. MLB, NBA, and NHL titles also supported saving to the HDD. Many other games utilized the drive to decrease load times or store additional maps.
There are a lot more Japanese releases that can utilize the HDD to reduce load times and store additional content or map packs. Many of these games appear to use very specific blocks of the hard drive space, ranging from 128MB to 1GB or more, to improve loading times. Since data would normally be read from a disk, which can be somewhat slow as the pickup head needs to move and scan the spinning disk, the HDD was a useful way to store and read data.