I would like to rent a car in Yerevan airport, drive it through Armenia, go to Georgia and leave the car in Tbilisi.
Are there any companies who offer that kind of service?
Hertz does, although I haven't tried myself. You can check Hertz's Armenian website (in English), where you'll also get the price and be able to book this exact service.
Hertz's Terms and Conditions even has a special section on renting in One Direction specifically for Georgia.
From my experience I will like to advise you NANIKO car rental in Armenia. You can rent a car in Yerevan and leave it in Tbilisi. Also as I compared early (5 month ago) NANIKO was cheapest, so highly recommended.
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