Tbilisi TV tower probably is the most visible construction in the city, located inMtatsminda  I haven't heard though it would be opened for public. 

Here is what we know about it:

Georgia Tbilisi TV Broadcasting Tower is a free-standing tower structure used for communications purposes. The tower is located in Tbilisi, Georgia and was built in 1972.

The preceding structure, built in 1955, was moved to the vicinity of the city of Gori.

Tbilisi TV tower

Tbilisi TV tower

The tower is operated by "Georgian Teleradiocenter", that was established 1955. Communication systems on the tower include regular broadcast, MMDS, pager and cellular, commercial TV, and amateur radio repeater. The tower is 274.5 m (901 ft) high on a mountain at 719.2 m (2,360 feet) above sea level.

The present tower has been put on the list of the unique European TV towers.