When I went to Hampton Court last weekend, I didn't have time to visit the palace but did manage to snap a few chimneys. (I say a few because there are 241 in total!)
These chimneys were an integral part of the palace's design - although many are now Victorian replicas, since the soft red brick is vulnerable to weathering. Indeed, conservation is ongoing (read about it in detail here [PDF]).
Why were chimneys so important to the Tudors? They were a response to the increasing use of coal, which creates more smoke than wood. The fanciness of these examples was a way of showing Cardinal Wolsey's, and later Henry VIII's, wealth. Spiralling and fluted designs also kept them cooler - important given the limitations of tudor bricks.