All right, so I just finished the game. I've got all 186 medals and I've got a gold star on every stage and every boss. Despite that, I'm still missing the final reward in the medal list. It says 186 medals, but I've got to assume I also need to complete the 35 challenges to unlock it.
http://wikiwiki.jp/kirby-ds2/?%A5%B3%A5%EC...%B7%A5%E7%A5%F3 says that the reward for the 186th medal is a survival sub-game.
I'm not sure how useful translating the challenge list from the game will be since some of the names are not at all descriptive. The link I'm translating it from has what looks like the requirements for them, though running it through google translate was iffy.
Challenge list translated from
http://wikiwiki.jp/kirby-ds2/?%A5%C1%A5%E3%A5%EC%A5%F3%A5%B8
CODEClear with 10 people
Delicious fruits
Pupon hunter
First 10 person clear
Golden enemy
First gold star
First turning into giant
All right with shaking!
With 10 people the Sand Queen
With 10 people the Forest King
Delicous Block
Perfect GOAL
Balloon Master
Can't be abducted!
Victory over rival alone
Bronze is no good!
Good at Tag
Star Rider
Catch Master
Panel Sniper
Pon Pon Bowler
With 10 people the Piggish boss
Reject heat
Nice Drive!
10 person clear 2
Let there be light! (more lit. Make it Bright)
Reject going back
Glittering space ship
Golden enemy 2
To the Goal at any time
Big things are good things
All Stage Clear
With 10 people the last enemy
Gold Star Master
Medal All Get