This list isn't guaranteed to be complete.
Final Fantasy X - Installs to the HDD to decrease load times. dsxgate supplied this information on the GameFAQs message boards.
Final Fantasy X International - Installs to the HDD to decrease load times. Takes up 1,664 megabytes.
Kingdom Hearts - Installs to the HDD to decrease load times. Takes up1,280 megabytes.
Xenosaga Episode 1 - Installs to the HDD to decrease load times. Allows you to save your game and load it to and from the HDD directly so that you don't need a memory card. Takes up 1,792 megabytes.
PlayOnline - Fully installs to the HDD so that it can be patched/updated. Used to boot Final Fantasy XI. Takes up 1,024 megabytes.
Tetra Master - Fully installs to the HDD so that it can be patched/updated. Comes with Final Fantasy XI. Takes up 128 megabytes.
JongHowLo - Fully installs to the HDD so that it can be patched/updated. Comes with Final Fantasy XI. Takes up 256 megabytes.
Final Fantasy XI (and the Rise of the Zilart and Chains of Promathia expansions) - Fully installs to the HDD so that it can be patched/updated. Takes up 8,192 megabytes as of March 7th, 2005.
Unlimited SaGa - Installs to the HDD to decrease load times. Takes up 3,072 megabytes.
Wild Arms 3 - dsxgate supplied this information on the GameFAQs message boards.
Ace Combat 4 - SScorpio supplied this information on the GameFAQs message boards.
Virtua Fighter 4 - SScorpio supplied this information on the GameFAQs message boards.
Armored Core 3
Taito's Energy Airforce - Allows you to playback music that is stored on the HDD while playing the game instead of listening to the music that is built into the game. gouki13 supplied this information on the GameFAQs message boards.
Kingdom Hearts Final Mix - dsxgate supplied this information on the GameFAQs message boards
Winning Eleven 6 Final Evolution - Nanami supplied this information on the GameFAQs message boards.
3D Fighting School 2 - Allows you to directly save to and load from the HDD. -
www.ncsx.com
Dark Chronicle (aka Dark Cloud 2) - Installs to the HDD to decrease load times. Takes up 1,536 megabytes. The Asia version also supports the HDD.
Capcom Vs. SNK 2 - Installs to the HDD to decrease load times and allows direct saving and loading to and from the HDD. - gouki13 supplied this information on the GameFAQs message boards.
Zettai Zetsumai Toshi (aka Disaster Report) - Installs to the HDD to decrease load times and allows direct saving and loading to and from the HDD. - gouki13 supplied this information on the GameFAQs message boards.
Soul Caliber II - Installs to the HDD to decrease load times. - I read about support on the back of the Japanese game case.
Star Ocean III: Till the End of Time - Installs to the HDD to decrease load times. Installation also reduces occurrences of a game crashing glitch that is known to happen on the first batch of discs when played on model 1x000 PS2s. - GameFAQs' Star Ocean III Message Board.
All-Star Pro Wrestling II - Markiss614 on the PS2Ownz.com forums.
Pop'n Taisen Puzzle-dama Online - Installation to the HDD is required to play. The game boots from PSBBN and does not require the disc or a registration code, making it a very unusual case of HDD support, as it has no anti-piracy protection to prevent the disc from being passed around in a group of people.
Net de Bomberman (aka Bomberman Online) - Installation to the HDD is required to play. Uses 1,024 MB on the HDD. The game boots from the disc, but loads most of the data from the HDD. A feega account is required to play, meaning you need PSBBN 0.30 or higher since you need it to access feega and make an account. The game has a monthly fee of a little over 500 yen.
SOCOM II (NTSC U/C version only) - Uses the HDD for three additional maps, which can only be installed from the discs that came with specific issues of OPM magazine.
The ESPN 2K5 sports series (all games) - Allows saving to the HDD instead of a Memory Card, and stores footage from gameplay for a halftime show with a movie instead of still images.
Nobunaga's Ambition Online (and its expansion) - Fully installs to the HDD so that it can be patched/updated.
King of Collisseum (both the Red and Green discs)
Front Mission Online - Fully installs to the HDD so that it can be patched/updated. Boots through PlayOnline Viewer once it is installed.