Some verbs, like 過ぎる can be added to the stem of other verbs to alter the meaning, (食べる>食べ過ぎる), can 過ぎる be used alone in specific circumstances? Are there verbs which are either grammatically incorrect when used alone, but still are classified as a verb, or verbs which can never be used alone in any context, despite being usable alone without being grammatically incorrect?
(Compiling a few comments into one big answer, hope nobody minds.)
To answer your first question, 過ぎる can be used alone. It means "pass," as noted in comments by user4092.
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Your second question is trickier. In modern linguistics, words are classified based on how they function. You could argue that a word that can't function as a verb really shouldn't be classified as one. (This is why many languages have a separate class for "auxiliary verbs", words that conjugate and otherwise behave like verbs but cannot serve as the main verb in a sentence. Some linguists see this distinction in Japanese too, and speak of an "auxiliary verb" class as described in comments.)
However, there are some words in Japanese that began as independent verbs but are now almost invariably used in an auxiliary-ish way (as part of a "compound verb phrase", if you like), like sobireru and aguneru as mentioned by chocolate. Many of these would probably fit the description "verbs which can never be used alone in any context, despite being usable alone without being grammatically incorrect," at least in contemporary Japanese. (People might concede the point if you argued that it wasn't wrong to use sobireru as an independent word, but it would probably still sound weird to them.)
Incidentally, looking at the history of Japanese, many morphemes that are now indisputably "auxiliaries," completely unable to stand alone, started this way (-masu, for example, was once mairasuru). Basically, what we are looking at is examples of verbs that picked up an auxiliary-ish use and then also lost their non-auxiliaryish use -- as opposed to verbs like "sugiru" that picked up an auxiliaryish use but still thrive as an independent verb too. (In fact "sugiru" has arguably split into an "auxiliary verb" and a "verb," depending how you want to slice things.)
i guess... Looking for a true [auxiliary verb] or verb-suffix in Jp.
How about めく ?
i don't think it can be used by itself.
－め・く --- 接尾語 カ行 四段活用
－そ・む 【初む】 〔動詞の連用形に付いて〕…し始める。初めて…する。「言ひそむ」「聞きそむ」「咲きそむ」「立ちそむ」