Here is the full list of all 45 Lil Big Invasion achievements.
Rescue all little fellows in World 1-4
Beat the Boss in World 1 and rescue all little fellows
Rescue all little fellows in World 1-3
Rescue all little fellows in World 1-2
Rescue all little fellows in World 1-1
Successfully complete the rescure training
Rescue all little fellows in World 1-5
Rescue all little fellows in World 1-6
Rescue all little fellows in World 1-7
Rescue all little fellows in World 2-1
Rescue all little fellows in World 2-2
Rescue all little fellows in World 2-3
Rescue all little fellows in World 2-4
Rescue all little fellows in World 2-5
Rescue all little fellows in World 2-6
Rescue all little fellows in World 2-7
Beat the Boss in World 2 and rescue all little fellows
Rescue all little fellows in World 3-1
Rescue all little fellows in World 3-2
Rescue all little fellows in World 3-3
Rescue all little fellows in World 3-4
Rescue all little fellows in World 3-5
Rescue all little fellows in World 3-6
Rescue all little fellows in World 3-7
Beat the Boss in World 3 and rescue all little fellows
Rescue all little fellows in World 4-1
Rescue all little fellows in World 4-2
Rescue all little fellows in World 4-3
Rescue all little fellows in World 4-4
Rescue all little fellows in World 4-5
Rescue all little fellows in World 4-6
Rescue all little fellows in World 4-7
Beat the Boss in World 4 and rescue all little fellows
Rescue all little fellows in World 5-1
Rescue all little fellows in World 5-2
Rescue all little fellows in World 5-3
Rescue all little fellows in World 5-4
Rescue all little fellows in World 5-5
Rescue all little fellows in World 5-6
Rescue all little fellows in World 5-7
Beat the Boss in World 5 and rescue all little fellows
Rescue 100 little fellows
Rescue 200 little fellows
Rescue 400 little fellows
Unlock Special World by beating the Game 100%