I don't clearly understand the meaning of "I have got __" in sentences like the following one:
I have got to get.
Is it a sentence using the past tense, or the present? What does it mean?
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It is an auxiliary verb with the the same meaning as "must". Here is what dictionary.com says:
auxiliary verb 2. Informal . must; have got (followed by an infinitive).
So, in your example, the meaning is "I must get..." Note, it is used with "have" which again is an auxiliary meaning "must", and I can't think of an example of where you could use "get" in this way without "have". But perhaps I am wrong about that. In this case, using both adds an emphasis to the importance of the "getting".
Note, as the dictionary says it is pretty informal, and should not be used in any type of formal communication.