Is this really how you would say "at the top of"? I just put "the guest is above on the mountain", which is not quite the same thing. (But Duo let me keep my hearts though!)
German doesn't have noun versions of 'top' and 'bottom', so it leads to what we, as English speakers, consider strange, clunky phrases. If you say "er ist auf dem Berg", it means that he is on the mountain, but to specify that he is on the top of the mountain you use 'oben auf...' I'm fairly sure that the bottom would be 'unter auf...'
Well, you could also say auf der Bergspitze which translates to on the mountain-top or peak of the mountain, or you have Baumspitze which is tree-top. How would you say "My name is on the top/bottom of the list"? What about "He is at the bottom of the picture", or "He is standing at the top of the stairs."
You could use Spitze/ Fuß for the stairs or the list as well. For the list also Anfang/ Ende. For a picture, you would most likely say "er ist unten auf dem Bild", which is still two words shorter than the English phrase, so you can't really call it "clunky" in comparison.
Oh, and Bergspitze = der Gipfel
Baumspitze = der Wipfel
British English allows "He is over (there) on the cricket pitch/field, waiting for the start of play." There is multiple redundancy in this, no doubt. You could equally say: He's on the pitch, waiting; He's over on the pitch, waiting, He's there on the pitch, waiting; He's over (there) at/by/next to the pitch, waiting, or even He's over there, waiting on the pitch. The combinations are bewildering, but I have heard them all and none of them seems unacceptable.
"oben" is an adverb, which stands alone and gives a static position "on top". "über" is a preposition, therefore precedes a noun phrase, with the meaning "over" above". The problem for English speakers is that the word "above" can be both. But when translating you simply have to check whether there is a noun phrase following or not.
The difference between "hinten" and "hinter" is exactly the same. "hinten" is the adverb ("Ich bin hinten."), "hinter the preposition ("Ich bin hinter der Tür".).
Sure. The main part of a "noun phrase" is a noun. It could be just that, so a single noun is definitely a noun phrase. But there could be lots of things added, such as articles and adjectives.
Examples: "Peter" is a noun phrase, and so is "that distant little mountain". "Peter's Grandmother" is a noun phrase as well. here "grandmother" is the main constituent, whereas the genitive attribute ""Peter's" is an addition. In extreme cases, additions can be complete relative clauses, as in "the boy I have seen yesterday". This forms a noun phrase, whose "kernel" is "boy".
I think ArtistryHM will still be confused after this. "oben auf dem berg" also contains a noun. But the point is that "oben" doesn't control the noun (it's a separate adverb here, describing the extend to which der gast ist auf dem berg. Meanwhile "auf" here controls "berg". So would "über".
You spotted the point! "oben" and "auf dem Berg" can be seen as standing beneath on another. You can leave out "oben" without rendering the sentence ungrammatical. You can even put a comma between "oben" and "auf dem Berg" or switch positions "auf dem Berg oben".
But "auf" is a preposition that controls, as you say, the noun phrase. You cannot leave it out (and it even directs the case of the following noun).
I disagree with the translation given by Duo. "Up on the mountain" rather than "on top of the mountain" is the translation that I would give. I am a native English speaker from Canada and Australia and the expression "on top of the mountain" is not used. Is "on top of the mountain" another "Americanism"? The only similar expression to "on top of the mountain" would be "at the top of the mountain" (auf dem Gipfel des Berges).