example of something that would raise gdp and yet be undesirable?

Answer 1

Allowing automakers to lower emissions standards -- price of cars would drop, production would increase, but pollution would also increase (undesireable).

Answer 2

Strip mining or other exploitative ventures contribute to GDP, but are generally considered to be negative. Another example would be the Three Gorges dam in China, which should contribute to the Chinese economy, but would widely be considered to be undesirable due to the widespread dislocation that is occuring of the people living in the valleys that are going to be flooded.
There are many economic activities that increase GDP, but have negative consequences that are either dificult to quantify or impact many people while only benefitting a few. This is widely refered to as the tragedy of the commons. This comes from the 18th century practice of allowing animals to be put to pasture on the public commons for free. As long as the numbers remain low, the commons are useful for everone. When individuals try to put more of their own animals on the commons, it has a negative consequence for everyone. As long as only one of two people graze more animals, they reap additional economic profits, but when many people do it, it simply ruins the commons for all. This is one of the occasions where government intervention can be positive, protecting the community from a few individuals who attempt to expropriate societal assets for themselves.

Answer 3

Cleaning up pollution.

Answer 4

war, natural disasters