Newgrange - Neolithic burial grounds at Newgrange are older than the Egyptian Pyramids, pre-dating Stonehenge by 1,000 years. One of Europe’s most important prehistoric clusters.

Perhaps the Lighthouse of Alexandria at an estimated height of 120 meters (423 ft).

The Jetavanaramaya in Sri Lanka, a Buddhist Stupa, is 400 feet tall. It was built between 270 and 301 AD. As the exact height of the Lighthouse of Alexandria is unknown, the Jetavanaramaya may have been taller.

The Temple Mount in Jerusalem

From http://www.jerusalem-insiders-guide.com/temple-of-jerusalem.html

The dimensions for the Temple of Jerusalem were staggering: 460 meters to the east, 315 m to the north, 280 m to the south, and the western wall was 485 meters long.

The walls above ground rose 30 meters (ten stories tall), and their foundations were as deep as 20 meters in some places in order to reach bedrock. Each layer of the wall was recessed about 3 centimeters from the layer beneath it. This was to avoid the optical illusion created whenever you look up a tall, straight object, that it is about to fall over you.

Some of the quarried stones used in the Western Wall are so large that, to this day, archaeologists have trouble understanding how they could possibly have been transported. The smallest stones weight between 2 to 5 tons and the largest stone of them all – possibly the largest building stone in antiquity – is 13.6 meters long, 4.6 meters thick and 3.3 meters high, and is estimated to weigh 570 tons. The builders used dry construction – there is no cement between the stones. In fact, there’s nothing holding the stones together except their own weight.

Today, the highest point in the exposed section reaches a height of 40 meters above the bedrock

What about the Pantheon in Rome, finished circa 126 AD?

It is 142 feet to the inside of its oculus, and the dome adds another 1.2 metres (3.9 ft).

(shared from engineeringrome.com via CC 3.0, attibuted to Lancaster, 2005).

Not only that, it is still is the world's largest unreinforced concrete dome, and the only one surviving from the time of the Empire.

Borsippa was a city that was closely connected to Babylon. It had a Ziggurat built by Nebbuchadnezzar II, on the site of an older building. It belonged to the god Nabu. Originally standing at 70 meters, the remains of the ziggurat are now 52 meters tall. Medieval people thought that it was the tower of Babel. Its known as the "the tongue tower" because of its distinct shape.

Dur Kurigalzu was a city during the Old Kassite period. It was named after King Kurigalzu, c. 1375 B.C., who made it the capitol of Babylon. Bricks on the ziggurat bear the inscription of Kurgalzu II, c. 1332 B.C. The remains of the ziggurat are about 52 meters tall. The surrounding platform was rebuilt in the 90's, and its a popular destination for people in Baghdad.

I'm not sure if the OP was asking about Mesopotamia, but it is technically a part of the Near East. There was nothing in Europe close to this size until the Roman Empire.

Possible currently existing candidates after the Colosseum and the Nimes Aqueduct.

at 91 feet, the aqueduct of Ferreres, tarragona, Ic B.C.

at 89 feet, the aqueduct of Segovia.

The Hercules lighttower, Galicia. 118 feet of roman building, 72 more feet added in the 1600's during a rebuild process.

a "detailed" plan of the roman stone structure ( no details added, no ornaments, no wood, only the stone plans as discovered):

The Alcántara bridge, 90 A.D. , 147ft height , thanks to @njuffa

I suppose that the Neolithic Silbury Hill in Wiltshire, built about 2470 - 2350 BC, is probably a contender for any list of the tallest European structures from ancient times.

At 39.3 metres (129 ft) high, I'm pretty sure it is the tallest prehistoric structure in Europe.

^{Photo by Greg O'Beirne, CC BY-SA 3.0, unchanged}