I have similar memories of my earlier times with FF7. The first playthrough is usually the hardest because you don't know where to find most enemy skills, but once you do, a ton of possibilities open up. Possibilities that you simply don't have with normal spells. Beta, Aqua Lung, and Trine where already mentioned, but the most overpowered ones in my mind aren't even attack spells. To me, they are:
1) White Wind: You can learn it as soon as you get the Buggy (Zemzeletts in Junon area have it, but you need Cait Sith's Manipulate materia in order to learn it), and at that time in the game it is about as powerful as a Cure2 spell, albeit a bit more expensive at 34 MP. However, it does have 2 major advantages over Cure2: first, the number of times you can cast Cure2 on the entire party per battle is limited by the level of your All materia, and for most of disc 1, that usually won't be any higher than 2, maybe 3. With White Wind you have a healing spell for the entire team that can be cast an infinite amount of times, as long as you have enough MP. Second, it also cures almost all status ailments! The only normal spell capable of doing that is Esuna, but it takes quite long to learn that one (I usually get it near the end of disc 2), and White Wind makes it obsolete from the get-go.
2) Angel's Whisper: this spell is basically Life2, Full Cure AND Esuna combined in one, but at only half the MP cost of either Full Cure or Life2! However, I guess the fact that you can only get it at the very end of the game makes this a moot point, but still... Life2 and Full Cure are also supposed to be end-game spells, so they shouldn't be outclassed like that...
3) Big Guard: Wall and Haste combined, and unlike either of them, the number of times you can cast it on the entire party is not limited by any All materia level. It's available at the same time as White Wind, long before you can buy even the standard Barrier and Magic Barrier spells! To top it off, it costs a little bit less than Wall. However, one might argue that Wall is overall better because the barriers last longer without haste. Still, I think it was a mistake to make a spell this powerful available so early in the game, would have been better if you got it near the end of disc 2.
4) Magic Hammer: Ethers just became completely obsolete? Free Magic recharge as long as your foe has MP left? With this, there's basically no more resource managing necessary in battle. Definitely needs to be nerved in some way. I'm thinking of the Magic Hammers in other FF games like FF5 or FF9 where it also damaged the enemy's MP, but didn't heal your own.
5) Bad Breath: thanks to this basically all green materia status spells become obsolete, especially since Bad Breath even has a higher success rate than most other status spells, if my memory serves right. It costs quite a lot of MP, but the ability to cripple most regular enemies and even some bosses makes it well worth the cost.
And those are just some of the overpowered enemy skills, with others like Magic Breath, Shadow Flare, and Pandora's Box deserving honorable mentions. But the best/worst part is: you get all of these spells for the price of just ONE materia slot! This makes it very hard for green materia spells to compete.