To me, 'typical American', depending on the context the two words are set, may be conceived as a non-phrase, as in 'He is a typical American', where 'American' is the head and 'typical' is an adjective that modifies the head of the NP; or as a group of adjectives, similar to 'beautiful big old' in 'beautiful big old car'.
Thanksgiving is a typically American holiday. -- typically is an adverb that qualifies 'American', an adjective
Thanksgiving is typically American. -- typically is an adverb that qualifies 'American', a(n) (predicate) adjective
EDIT: Compound adjectives comprise a participle FORM, present or past. 'American' is not a form resembling to a participle --> no participial adjective or compound adjective.
I hope my comments are helpful in some way to you.