It does not sound very good and native speakers tend not to say "extremely great". This is because "great" is usually an "ungradable" adjective. This means that its meaning is already very strong or extreme and so it sounds excessive or ambiguous to then use adverbs like "very" or "extremely" with it. "Great" means "very good" and so extremely great means "extremely very good"! You can use words like "absolutely", "totally", "completely" with "great" and other ungradable adjectives. You can use "really" with any adjective - it's a really great option if you can't think of anything else!

It is not wrong, but we don't really say it. Extremely, is a very strong adverb and great is a strong adjective, so you wouldn't really put them together.

I guess it's alright, but I think that "extremely good" or "very good" sounds a bit more natural.