Hyperbole

Puke. Noun. It's a naming word.

The phrase is "towards (the condition of) nausea". Accusative of nausea is nauseam. I don't get why you're struggling with this, since you even managed to provide your own link to the correct spelling and translation.

TO puke. TO TO TO puke. Removing the "to" to change the meaning doesn't really get you anywhere.

ad nauseamadverb (Latin) again and again, over and over (again), on and on, time and (time) again, time after time, ad infinitum, times without number We discussed it ad nauseum.
Collins Thesaurus of the English Language – Complete and Unabridged 2nd Edition. 2002 © HarperCollins Publishers 1995, 2002

Accusative?

Simply asserting by repeating is not proof. Argumentum ad baculum.