For games primarily about mountain (etc.) climbing. Mountain climbing as genre, basically. Over half of these are PlayStation games.

Chronology

• Cabela's Survival: Shadows of Katmai (Assassin's Creed-grade climbing though)
• Climb play
• Climbing Flail
• Everest
• Everest: Iceman's Apprentice
• Extreme Rock Climbing (though it's indoor rock climbing, rather than mountain climbing) info.
• Final Assault
• GIRP
• Grow Home
• Mitouhou e no Chousen: Alps Hen | 未踏峰への挑戦 アルプス編
• Mountainer (More of an endless runner/dodger that happened to have a mountain background maybe.)
• Mt. / Mount (mountains are generated from words entered by player)
• Rock N’ Roll Climber (kinda a Wii-ified GIRP)
• Solo Joe
• Souten no Shiroki Kami no Za: Great Peak | 蒼天の白き神の座　－ＧＲＥＡＴ　ＰＥＡＫ－
• Trepidatious Peak
• Power Climb

• 2022 - Climber: Sky is the Limit get

Games that use mountains as a setting, but do not feature mountain climbing as a primary mechanic.

• Aconcagua - HG101 article
• Agartha - unreleased
• Cursed Mountain
• Logic Puzzle Rainbow Town (for one set of nonograms)
• The Tozan RPG | SIMPLE1500シリーズ Vol.92 THE 登山RPG ～銀嶺の覇者～

Games that include mountain climbing sections.

• Rocky Mountain Sports / Mountain Sports (unreleased SNES / Jaguar)
• Terranigma - Players obtain a claw item that enables the climbing of cliff faces. It feels nice.

• K2

• standing_on_a_cliff
• Mountain climbing simulations - Gamengai forum thread.
• http://www.ne.jp/asahi/gamo/yama/data/game/game.htm - Mountain climbing games, on what I believe is a general mountain climbing page.