I would suggest: I climbed a hundred flights of stairs today, so I'm pretty exhausted. (Use the Simple Past if you're thinking of the day as a completed time period.) OR I've climbed a hundred flights of stairs today, so I'm pretty exhausted. (Use the Present Perfect if you're thinking of the day as a time period that you're still in.)

I agree with Jonathon. When 'exhaust' is a verb it is usually transitive and needs a direct object: (I exhausted all other options before taking out a high interest loan; The company exhauseted all its resources, etc.). It is usually used in the past tense as an adjective when talking about how one feels. So, I FEEL exhausted; I WAS exhausted, etc. Not just I exhausted.