You don't need a taxi to visit the top of Lycabettus Hill. There is an underground Lycabettus Funicular from the Kolonaki district of Athens since 1965. The trip is fast and relaxing. Its entrance is at the intersection Ploutarchou street and Aristipou street.
Taxi drivers will take your money, they will drive you to the much lower altitude of Lycabettus Theater parking and you will be forced to climb for half an hour the stairs to the top of Lycabettus.
The only bad thing is that from Kolonaki district and generally in the whole area there aren't signs for the Lycabettus Funicular. Check google maps, Wikipedia or anything else you like.
On the top of Lycabettus is the Agios Georgios church Saint George in English. The view is amazing. The hill is higher than acropolis and is the best place to view the Athens. You will realize that Athens is a pit surrounded by mountains. That prevents the strong winter winds and gives to Athens a mild climate.
Near the church is a coffee shop restaurant. Not the best restaurant but with the best view. Prefer the sunset it is a fantastic place to watch it and don't forget your camera.
If you don't want to pay for the descent of Lycabettus Funicular, try the stairs to the Kolonaki district and not to the Lycabettus Theater. Of course descent is by far easier and the route is under trees. Avoid the night hours though. Daytime is very safe.
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