In season two we see that Riker has grown a new beard. This is widely regarded as the turning point of the show into being a mature sci-fi, and even helped coin the phrase 'growing the beard'.
What prompted the actor to grow his beard?
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According to Jonathan Frakes (the actor who portrays Riker) the beard was a result of his personal hatred of shaving, combined with the 1988 writer's strike meaning that he had more time between shows than normal. When he attended a script meeting after the strike ended, Roddenberry said that he liked its "nautical" connotations and it just stuck.
Q: The beard, was it just a thing you did?
Frakes: What happened was there was a strike, there was a writer's strike after the first season, then we had a meeting with the cast and the late, great Gene Roddenberry, Berman and others. I had grown a beard because I hated to shave, and Roddeberry 'in his vision' said "Jonathan, I love the beard. It'll be nautical. But we'll shape it, and that way it'll be decorative".
In TNG: The Pegasus, Riker explains that he was sick of hearing people comment on his youthful looks.
PRESSMAN: How long have you had the beard?
RIKER: About four years. I think I just got tired of hearing how "young" I looked.
Although this is commonly referred to as a retcon, there's at least one instance of Riker being called 'young' in the first season, by Troi in TNG: Haven. He may simply have been brooding on this incident for a while.
RIKER: Have you discussed this with Wyatt? I think you should. It's also damned unfair to me.
TROI: I understand. I should have realised. Humans, young human males particularly, have difficulty separating platonic love and physical love.
RIKER: The problem is, Imzadi, I couldn't. Not now. Call it an old Earth tradition, habit of the beasts, whatever.
In Pegasus, his former captain reminds him of his former nickname, "Ensign Babyface". He just felt he had a young face and decided to grow the beard.
Further info from Memory Alpha page on Riker:
Jonathan Frakes grew the beard during the hiatus before TNG's second season as he dislikes shaving. He returned to rehearsal before shaving it. The producers liked the change and asked him to keep the beard, although a fictional reason was not given until five years later.