Unlike the Hanging Gardens, there is archaeological evidence at the dig-site of Babylon to support that Nebuchadnezzar II commissioned the Tower of Babel. You can see its remnants here.
Only it wasn’t a place of babbling gibberish that was destroyed by God. It was a ziggurat named Etemenanki, a temple to the god Marduk that was later destroyed by Alexander the Great. He wished to rebuild it in his image, but died before that could happen. Many people subsequently tried to rebuild it in their own images, each time tearing down what had been rebuilt to start anew. But nobody ever finished it. It looks like this place ended up representing humankind’s inability to work together after all.