Tower of Pisa
Even before the tower was finished, it started to lean eastwards.  You can see that the top piece is at a different angle to the tower as the builders tried to compensate.  It is relatively stable in that it stayed this way for 800 years, but it was slowly still tipping, and the estimate was that in about 200 more years it would topple.  So, engineers have excavated under it on one side and placed weights in it to bring it back up slightly.  While I was visiting, the tower was off limits due to this work.  It re-opened in about 2000 in it's now more stable configuration.