These are roughly listed in chronological order.
LEGO Island 2: During the time that I was 7-9 years old, every bit of time I spent playing this game was playing the minigames over and over again in an attempt to get gold on everything. After a year to a year-and-a-half spent working towards this goal, I FINALLY got gold on every last minigame and island, and could go to my shelf of trophies and see nothing but gold.
Rayman 3: After many months of sharpening my score on all the levels and doing things to get the maximum combo, I got my score really high.
Rayman 2: It took a very long time, but I eventually found every last yellow lum.
Rayman Arena: Beat every last area there is, and also beat Speed Stress. Just so you know what this is, you were chased by a giant wave and had to run over acceleration pads and repeatedly tap a button that would give you an extra boost if you did so. Surviving needed some very rapid button presses along with quick arrow key presses that made my brother laugh pretty hard at seeing my hands move with such inhuman speed, and hearing the keyboard buttons being mashed so rapidly. My dad just so happened to be around during this, and he cringed and threatened to take the keyboard away due to concerns regarding the health of my hands. (despite this coming up on several other occasions, he never followed through with it)
Sonic Heroes: When you first complete the story mode with a team, the ending screen shows all the characters from the game and says something like: "Try another team." Then when you finally beat the story mode with all four teams, the ending screen shows the seven chaos emeralds and says something like: "Collect all seven chaos emeralds." Then if you do that, the ending screen shows metal Sonic and says: "Defeat last evil." Then when you do the extra final boss, the ending screen says: "Get all A rank." So then when you finally go to all the levels and get A on every single one of them, the ending screen says: "Try SUPER HARD mode." And then, when you play super hard mode, it's basically the story mode with Team Sonic except all the levels are harder, and when you beat it, finally, after all that hard work, the ending screen says: "Thank you for playing." Doing all this stuff is pretty hard, and extremely time-consuming. I had a lot of free time when I was ten, and even then it took me 81 days from the day I got that game to get to that ending screen, and my brother was completely left in the dust and never got all A's in his file.
Halo: Beat the campaign mode on all four difficulties.
Halo 2: Beat the campaign mode on all four difficulties. (all four difficulties are harder in Halo 2 than they were in the first one)
Cave Story: Fell in love with the game in less than twelve hours, and by the time one month had elapsed, I had played through the whole thing at least ten times, and seen every last bit of dialogue that it's possible to see in the game to the point where the only bits that were new to me when I looked through the code were those that never execute unless the game has been modified. Additionally, within this one-month timespan, I got a sacred grounds time under three minutes and got to see Sue at the main menu. In the following months, I kept polishing my time, and for some time held the fastest known Sacred Grounds time with the Spur. While I have since been surpassed, I plan on reclaiming my title.