In a manner that causes extreme tiredness or uses up one's energy completely; draining or depleting.
From exhausting (present participle of exhaust used as an adjective) plus the adverbial suffix -ly. Standard English formation from the 19th century.
The word exhaustingly describes both the experience (you're tired) and the cause (something is draining)—it's like the language is acknowledging both the victim and the culprit simultaneously!
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