great good luck or greatly lucky for using in a sentence ... | great good luck
"Greatgoodluck"isanidiomaticphrasethatsrarelyused."Great"doesnotmodify"good".Itsactuallyjusttwoadjectives,bothmodifying"luck".Otherexamples:AlargeroundstoneAriperedtomatosmoothcreamybutter"Greatly"isanadverbthatisnotuncommon,butnotidiomaticallypairedwith"lucky".Thereisnoreasonforthis,itsjustnotsomethingnativeEnglishspeakerssay.However,asP.E.Dantmentionsinhiscomment,"extremelylucky"and"hugelylucky"canbeidiomatic.
"Great good luck" is an idiomatic phrase thats rarely used. "Great" does not modify "good". Its actually just two adjectives, both modifying "luck". Other examples:
A large round stone
A ripe red tomato
smooth creamy butter
"Greatly" is an adverb that is not uncommon, but not idiomatically paired with "lucky". There is no reason for this, its just not something native English speakers say. However, as P.E. Dant mentions in his comment, "extremely lucky" and "hugely lucky" can be idiomatic.