Challenge Times are special level completion times that the developers put in for each level. They are hard to beat, but give you Coins Galore stages. To unlock the Challenge Times, simply collect all gems and complete all Bonus Challenges in an episode.
Crown for Captain Toad to Wear
After doing all three episodes of the main game and completing every stage, finding every gem, and doing every extra challenges the levels offer, the bonus book pages become available. After 100%ing those, Mummy Me Maze Forever becomes available. Get to the 50th floor on that and you unlock a crown for Captain Toad to wear in any level.
Crown Toad and Time Trials
Complete Mummy-Me Maze Forever.
Hidden Title Levels
Leave the game on the title screen for 30 seconds and it will load either a fountain level or a gallery level. Walk up to the blue toad for a mushroom worth 10 coins.
Mummy-Me Maze Forever
Obtain every Gem in the game and complete every extra objective in the game.
Prologue Levels
In order to unlock the secret levels in the bonus book, you must complete the book that the level is in. Ex. Complete book 1 = prologue level 1
Super Mario 3D World levels
Have a saved game from of Super Mario 3D World. A new option will appear at the main menu, that allows some levels from that game to be explored as Captain Toad.
Toad Stamps
There are a total of three Toad Stamps on the front cover of each Episode book. Successfully complete the indicated task to unlock the corresponding stamp.
First Toad Stamp: Successfully complete the episode.
Glittering Stars: Have all three Toad Stamps, then beat the Challenge Times for each level.
Second Toad Stamp: Collect all gems in the episode.
Third Toad Stamp: Successfully complete all Bonus Challenges and collect all gems in the episode.