Generally considered to be a variation from どいかい (doikai), from the emphasizing prefix ど (do-, “very”) and the adjective 厳い (ikai, “great, huge, many”). The Nihon Kokugo Daijiten states that the sound change of /doikai/ → /dekai/ does not match the facts of the term's development. However, the /oi/ → /e/ sound shift is quite common in Japanese, and the question may hinge on which form appeared first, でかい (dekai) or でっかい (dekkai). If でかい (dekai) evolved from でっかい (dekkai) instead of the other way around, the do + ikai derivation would not explain the geminate /-kk-/ in でっかい (dekkai). The Nihon Kokugo Daijiten suggests an alternative derivation of でこ (deko, “hugely, enormously”, adverb; old slang or dialect term, persisting in modern でこでこ (dekodeko)) + 厳い (ikai, “great, huge, many”): */deko ikai/ → */dekoikai/ → */dekekai/ → /dekkai/ → /dekai/.
Also said to be the adjective form of でかし, from いでかす, transitive of 出で来 (ideku, “to appear”). However, the semantic development required for this is improbable, and this is more likely a folk etymology.
でかい (-i inflection, rōmaji dekai)