Sure, you could say "I already have that book" or "I am already here"

Using already is to prove a point, you may say it in a positive/negative way to define the context. Husband: Is the food ready? Wife: Yes it's already on the table. Cop: Did you put the phone on the table? Suspect: No, it was already there.

Thanks for your answers :) So using 'already' in a present tense construction makes sense even though it implies an act done before the moment you're in or earlier. What caused me to ask this question is that I came to see a writing of someone as in 'Already all language(s) take words from each other' which made me think whether this is a correct structure or not. What do you think of it? If it's correctly said, what does it exactly mean?

Yes, you can say something like "I am already doing that."