I am not sure if this sentence is correct
La marche m'a tellement fatigué !
The verb "se fatiguer" is not pronominal? Is "fatigué" an adjective?
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Yes, the sentence is correct.
The confusion comes from both forms existing : se fatiguer (pronominal) and fatiguer [+ facultative object].
Some examples :
Pronominal form : Quand je cours, je me fatigue.
Non-pronominal (yet transitive) form : Les efforts fatiguent l'organisme.
Non-pronominal (AND intransitive) form : La marche, ça fatigue.
Here, the subject of the sentence (La marche) is not the same as the object (moi, i.e. the locutor, in the contracted form m'), so it can't be the pronominal form of the verb here, but it's correct and meaningful nonetheless.
Additionnaly, to answer your last question, where you wonder if it could be an adjective, you can resolve this type of case quite easily, by switching said supposed adjective with another one (but not based on a verb), to check if it's valid (grammar-wise, not meaning-wise).
« La marche m'a tellement grand. » (incorrect)
« La marche m'a tellement méchant. » (incorrect)
It's a good way to tell that in the original sentence fatigué can't be an adjective.