Question: what does "ichiban genki" mean in Japanese? 

Answer: i'll try to explain. ichiban = number 1, the best, the most. genki is a bit harder to explain. in the sentence, "genki desu ka?", you are asking somebody if they are okay, fine, in good condition. the reply to that is "genki desu", "i'm fine". it also means, energetic or playful. example: that girl is so genki!" in some ways, it also implies cuteness. like in manga/anime, it's not impossible to find a story with a title like "the adventures of the genki boy/girl" do you get it? so i guess your friend's cat is a burst of energy, hence s/he got that monicker, which i think is cute.
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