I didn't actually end up finishing a lot of games when I was a kid...
Battletech: Crescent Hawk's Inception
Not the hardest game in hindsight considering much of the world is empty, but finally figuring it out without any help as a kid (blowing a hole in the arena to escape) seemed like quite a achievement at the time.
Ultima
Not sure if this is hardcore or not. At once point 9-10 years ago I decided to try going through all the Ultima games in order. This was actually a ton of fun to play, I really got into it. Drew maps, printed out the world maps and wrote all over them, took notes, everything. I think as a kid I would probably have lost interest and never figured this out (my first Ultima was III btw). I can't say this game was difficult, but figuring it out felt like a bit of a puzzle. I felt that the most important thing was to get an overall strategy going at first to get powerful quickly (for me, this was finding a dungeon near a town and mapping it out as I fought, since you used way less food [if any?] in dungeons).
Final Fantasy 1-3
More recently, on NES/Famicom. Never finished any of these at the time, and yes, they are trivial if you grind (1 or 3) or take advantage of system quirks (2). I played them straight with no grinding, just buying what I could afford and dealing with it, and had a great time. FF1 is the hardest to play this way, while 2 and 3 are balanced so that (as long as you understand the systems) you can get through without a problem. However, even for those two, if you pay no attention to the system you'll get hopelessly stuck, so I felt quite clever for playing them straight to the end.
Demon's Souls/Dark Souls 1 & 2
I don't even consider these to the RPGs since they are 100% player skill based, but I seem to be in the minority with this belief. These games are not that hard, and grinding really doesn't do much in my opinion. Getting good at dodging and anticipating attacks gets you through all games, and they all even have "easy modes" (magician/pyromancer). I played online, but never summoned any help nor used any NPCs, and I felt pretty satisfied when each game was over.
Fallout
Is this even "hardcore?" I actually found it to be easy and overall it was a pleasant experience. I bought the game when it first came out and never got very far into it for lack of interest. A year ago I had some free time and got addicted, getting through the game in a week or so of sporadic play. Awesome game, but I don't really feel like I had to figure anything out or really struggle to get through it.
Dragon Warrior
Unlike its sequel, DW is more tedious than hardcore. There is little room for tactics/strategy (basically none). It is just a fancy slot machine. You figure out how likely you are to survive at a certain level based on the avg damage you give/take and your HPs, and push your luck just to the point where you can barely get back to town in one piece. That's not bad. The bad thing is, you will definitely get to a point where you simply are not capable of giving/taking enough damage to get to the next point, so you have to sit there and gain levels. For me it was two spots (roughly the middle, then during the last are). With only one character there's only so much you can do, but I felt like Ultima was overall a much more clever game since most of your power came from equipment and knowledge instead of your level and HP.