Here is the full list of all 22 reky achievements.
Be the best in 30 levels
Be the best in 40 levels
Be the best in 50 levels
Be the best in 60 levels
Be the best in 70 levels
Be the best in 80 levels
Be the best in 90 levels
Be the best in 95 levels
Achieve 5 high scores in a row.
Achieve 10 high scores in a row.
Achieve 15 high scores in a row.
Achieve 20 high scores in a row.
Achieve 25 high scores in a row.
Achieve 35 high scores in a row.
Achieve 45 high scores in a row.
Explore Timidly
Explore Curiously
Explore Bravely
Explore Masterfully
Climb!
Climb Again!
Climb One Last Time!