Well, even as a native Chinese speaker, I don't know why we repeat verbs in certain cases. It's just... we do. It seems to me that 我看看 sounds more casual and approachable than 我看， but we don't repeat adjectives as often as verbs when speaking Chinese. For example, we don't say 他高高了. If we want to say "he is very tall" in Chinese, we add adverbs. In this case, we say "他太高了“. Since I'm not a professional Chinese teacher, these are only my opinions based on my experience of using Chinese. Hope there would be more professional explanations for your question.

to express is quick, not for long time

Very good question. Chinese is very complicated, in my opinion, the same verb will be repeated in these below situation 1. To express one dynamic process , like 我看看 or 我听听 or 我想想 , which means you are watching or listening or thinking .2. Follow the adjective, to express the stronger emotion. Like 亮晶晶 or 白花花 or 冷冰冰. 3. the repeated verb is lovely sometimes, like 吃饭饭 or 我怕怕.

我觉得，因为“看”是动词，‘高’是形容词。“我看”，这个句子不完整，“我看看”利用重复，来省略动词后面的内容，比如“我跳跳”“我瞅瞅”“我想想”。“我看”是不可以作为完整句子的，但是“他高”可以。