My first batch of games was Mario, Mario Land 2 and Yoshi. later i got Pokemon Red....then like 10+ years later, i got Azure Dreams and Wario Land.
New games for the family was quite rare outside of Christmases and Birthdays in the 90's, nowour days, we have money to get games whenrever or the internet to look into aswell, theres probally a decent auction in the UK somewhere.
Dragon Warrior (aka Dragon Quest) Monsters 1 and 2 were pretty decent games, you get monsters and then raise them, but it's 3 on 3, you can also breed monsters together to get a stronger or different type.
Azure Dreams felt a bit frustraiting and extremely grindy, you're this kid who has to look for his father for some reason, so you storm the tower in the middle of the city, but it's swarming with monsters, you get one monster to start and you have to find eggs and hatch them, but even then, your party is only as large as the amount of collars you have. In that game, you can fuse monsters aswell though i think all it does is boost the stats or changes its alignment or somthing strange, it is a dungeon crawler with all the traditional things, such as traps that makes weapons and armor rusty, theres even an item that turns a monster into an egg....espcially useful for one monster that protects itself by turning thrown items into other items whilst expending mp, it's a real gamebreaker.
The Final Fantasy Legend trilogy is nice aswell, though the 1st one felt a bit archaic and the 2nd one a bit confusing, the 3rd one is considered as the best, the main gimmick is that consuming meat from monsters or parts from cyborgs/mechanical monsters would usually change your character's forms into other monsters from the game.
Final Fantasy Adventure is sort of a hack and slash game, it was a bit so-so, but it was alright, can't say if it was memorable to me or what. (it is really part of the Mana series, it was remade on the GBA as Sword of Mana).
Legend of Zelda: Oracle of Ages and Oracle of Seasons were awesome games, they were games outsourced to Capcom and they did a pretty good job of producing them, they felt like an improvement over Link's Awakening, some reason though i like Seasons more than Ages, and once you complete one you can either connect it to the other or copy down a password so you can get bonuses.
EDIT: Forgot to menction that Oracle of Ages and Seasons can only be played on a gameboy colour or better only.