Just to see what one can do with skills, I built an expert mountain climber with this concept as the skeleton I built off from:
Sir George Mallory, with assistance from his trusty climbing partner Sir Andrew Irvine, and their awakened shirpa gorrilla, Coco (who has a climb speed).
Sir Mallory is the last of the Renaissance Men! He is proficient in all 18 skills (and many tools), has expertise in 10 of those 18 skills, and has further expertise in Intelligence or Wisdom checks relating to Mountains. He's also a Jack of All Trades, adding half his proficiency bonus (+3) to any ability checks he's not already proficient in, including tools, initiative, and general checks not associated with a skill. He also is a fine spellcaster (13th level spell slots, 4 spells from any list through Magical Secrets). He's strong as an ox, but not too wise (because mountain climber ).
Sir George Mallory (Bard 10, Rogue 8, Ranger 1, Cleric 1, 20th level total)
Str 20, Dex 15, Con 14, Int 12, Wis 8, Chr 12
Half-Elf (2 Skills: Medicine, Acrobatics)
Background: Mountaineer; Skills: Athletics, Survival; Tools: Climber's Kit, Herbalis Kit
Rogue 8 (4 skills: Perception, Insight, Stealth, Investigation)(Expertise 4 skills: Perception, Insight, Stealth, Investigation)(2 Ability Boosts, both to Str, to bring it to 20)
Ranger 1 (1 skills: Animal Handling)(Natural Explorer: Mountains; Expertise when making an Intelligence and Wisdom check in a proficient skill, expertise if it relates to your favored terrain)
Cleric 1 (2 Skills: History, Nature)(Expertise: History, Nature)(Guidance Cantrip)
Bard 10 (1 Skill:Arcana) (Lore: 3 skills: Slieght of Hand, Intimidation, Performance)(Expertise 4 skills: Athletics, Survival, Acrobatics, Animal Handling)(Magical Secrets 4 spells from any other list) (1 Ability Boost, 2 Feats)
Feat 1 = Skilled (3 skills or tools: Religion, Thieves Tools, Disguise Kit)
Feat 2 = Athelete (Dex/Str +1, Climbing doesn't halve your speed, better at jumping and standing from prone)
Expertise: Athletics, Acrobatics, Stealth, History, Investigation, Nature, Animal Handling, Insight, Perception, Survival (he could actually switch this to just Str, Dex, and Charisma skills, so that when he is on the mountain he uses his Ranger natural explorer ability to apply expertise on all those Intelligence and Wisdom checks which relate to the Mountain and therefore have expertise on all skills, but that seems excessive).
Now Sir George could of course cast Spider Climb, and just climb up the mountain wih a climb speed. But that would not be sporting! He's going to do it the old fashioned way.
He's not, perhaps, above some minor spellcasting to help him out, however. So he is willing to have Guidance cast on him sometimes by his partner (+1d4 to skill check), and he will cast Enhance Ability on himself (Bull's Strength = Advantage on Strength Checks, doubles carrying capacity).
He climbs mount Neverest at full speed (thanks to the Athelete feat), easily jumping over obsticles, climbing over them, swimming them, squeezing through them, hanging from them, etc..
He makes his Climb Check at advantage (Bull's Strength) with a bonus of +17 (5 Strength, 6 proficiency, 6 expertise (Athletics)) +1D4 (guidance) for an average climb check of 29.5 (plus advantage, so really more like 34.5). Any time he makes an intelligence or wisdom check relating to the mountain, he does it with expertise as well, even if it's a skill he didn't previously have expertise in.
There are of course a number of magic items he could use to aid in this quest. Gloves of Swimming and Climbing would add an additional +5 to the climb checks, for instance. And if anyone would have such an item, uncommon though it may be, you'd think it would be our Sir Mallory. So his climb checks are more likely to be 34.5 (or with advantage an average of 39.5).
A Nearly Impossible skill check is a 30. He should be fine.