Is there a word that specifically refers to reaching the top of a hill or mountain?
I want to describe the action of reaching the top of a hill and finally being able to see what is on the other side. Sorry if my explanation is not clear enough.
You could use summit as suggested by John D, or you could use "crest".
Whether you use summit or crest could be influenced by whether you are climbing a mountain or walking up a hill. It is more usual to talk about the summit of a mountain, and the crest of a hill (compare the previous reference with this).
"We crested the hill..." or "We summited the mountain..." .
I think crest would be better for the context you describe. "Summiting" tends to suggests that getting to the top was the ultimate goal, whereas cresting does not seem to suggest that.
(meta: This is my first time here, so be gentle. ... respectfully,)
'summit' and 'summited,' are the terms I've heard most frequently employed by actual mountain climbers.
MW Online, 2summit intransitive verb
2 : to climb to the summit
summited on May 29
Also, I've heard the intransitive verb 'peaked' used on a few occasions, though far less frequently. Rightfully or not, the term 'crest' carries connotations of wave-like fluidity and effortless grace. Whereas 'crest' sounds appropriate for ascending 'hills,' which cannot unreasonably be considered small mountains, the word sounds ill-fitting when one is describing the effort and danger involved in climbing mountains (which are hills on steroids). While I agree that the dictionary supports the use of the term 'surmount,' I find it rather archaic. As to its adversarial connotations those who scale true mountains consider themselves as contestants in a deadly sport and routinely anthropomorphize their subject.
(meta: Excellent question, link64. Please excuse my verbosity, I haven't had an opportunity to exercise my passion for a number of years, and then there's the medicine.)
mountains get conquered. hills get 'climbed'.