Ignoring other forms of magic like Thu'um, Sword-Singing etc. Here are my thoughts.
Sotha Sil might have become a god but he was always a powerful mage and I think even without his divinity he would have been one of the most powerful today. Close to four-thousand years old at the time of his death he constructed the Clockwork City, helped fight and banish Mehrunes Dagon from Tamriel and was at one time a member of the Psijic Order.
Dagoth Ur like Sotha Sil he became quasi-Divine but he was reputed to be a powerful mage before the events of the Heart. From Vivec's own mouth he is described as being "older than we, and a mighty sorcerer and enchanted in life".
Phynaster, now worshipped as a god/ancestor was an Aldmer/Altmer who lived in the Merethic or First Era. His most notable feats were teaching the Altmer how to extend their lives by one hundred years and creating the "Ring of Phynaster".
Shalidor was a First Era mage. His feats are likely to be largely exaggerated but regardless his was no doubt exceptionally powerful. A quote from the book, A Minor Maze, "Various tales tell of him battling Dwemer legions single-handedly, building the city of Winterhold with a whispered spell, stealing the secret of life from Akatosh, or constructing Labyrinthian himself".
Kagrenac was the chief dwemer mage-architect and was responsible for creating "Kagrenac's Tools" and later Numidium.
Orgnum is the king of the maormers and is described in a book as being "a deathless wizard". He is supposedly old enough to have lived in Aldmeris which would put him at over 5,000 years old and likely the oldest on this list.
Mannimarco is widely considered to be the "first lich" in history. I think it's debatable but it doesn't really matter. He's over 1,100 years old and is easily the most famous necromancer in Tamriel.
Divayth Fyr is the oldest Telvanni mage. He is "a 4000-year-old wizard" (as of 3E 375) and his daughter/clone refers to him as being "probably the oldest and most powerful wizard alive. Not counting liches, or divine sorcerers like Vivec, of course. And there may be some older in the West, on Summerset Isles, perhaps". And yes, he can create clones of himself.
I'm a huge fan of Ahzidal. He's older than Divayth and has knowledge of "ancient runes and dawn-magic even the elves had begun to forget". Too bad they just turned him into another generic dungeon boss in Skyrim.
Iachesis was the leader of the Psijic Order and older than Divayth Fyr, born in the very earlier First Era, so I'm sure he is powerful. The last time we hear of him is in 3E 110 so he is likely still alive.
The story of Hasedoki was a good one (you can read about it by looking up the Staff of Hasedoki). Hasedoki was a wondering mage who challenged other mages to duels. Apparently he never found anyone who could best him. That doesn't mean he is necessarily the most powerful but worth a mention at least.
Like Ahzidal, I have always been disappointed with how Bethesda implemented the King of Miscarcand. This ancient Ayleid king lich who should be close to, or even surpassing, Divayth's age is treated no differently than any other lich you find in a cave. No dialogue and no real point. Just a waste.
Azra Nightwielder was the creator of Shadow Magic. Probably not deserving of being on this list but I think he deserves a mention for his discovery of Shadow Magic.
I was never been impressed by Jagar Tharn, I mean what exactly is it that warrants his place as one of the most powerful mages? Infamous, cunning and certainly powerful he might be but he never struck me as being anything more than "Mages Guild Archmage level".
Mankar Camoran might been instrumental in assisting Dagon with his invasion of Tamriel but I question whether or not it was anything that any other archmage could not do. I think it would come down to whether or not he actually created his Paradise or whether it was largely Dagon's doing (from memory there is something about it actually just being part of Dagon's realm).
I've always considered the legacy Vanus Galerion left behind (the Mages Guild) to have led people to exaggerate his strength as a mage. I think Mannimarco is significantly stronger (since he is significantly older). He probably misses out on making the list.