One who encounters the last parts of the answers to the starred clues on the way up Crossword Clue

 The answer to the crossword clue "One who encounters the last parts of the answers to the starred clues on the way up" is MOUNTAINCLIMBER.

This is a 10-letter word that refers to a person who climbs mountains. The clue is a bit of a pun, as it suggests that the mountain climber is literally encountering the last parts of the answers to the starred clues as they ascend the mountain.

The starred clues in the New York Times crossword puzzle are the ones that are worth double points. The answer to each starred clue is made up of two words, and the last part of the first word is the same as the first part of the second word. So, for example, one of the starred clues might be "DOWN TRAIL". The answer to this clue would be "TRACK DOWN".

A mountain climber would encounter the last parts of the answers to the starred clues as they ascended the mountain, because the starred clues are typically located towards the top of the puzzle. So, as the mountain climber gets higher and higher, they would be encountering the last parts of the answers to the starred clues.

I hope this helps! Let me know if you have any other questions.