TITLE: Non-Noetherian ring with a single prime ideal
QUESTION [14 upvotes]: What are the most simple examples of a commutative ring $R$ satisfying both of the following two properties:
  1. $R$ is not Noetherian.
  2. $R$ has exactly one prime ideal.

REPLY [3 votes]: I thought this observation might be useful to future viewers other than the OP. If a mod thinks this is old enough for a new topic, please do.
Since the nilradical is  in every prime, the condition of having only one prime ideal (which will also be maximal) is equivalent to the condition that the nilradical is a maximal ideal.
Thus one immediately obtains a large class of examples as mentioned by the posters above as non noetherian rings of the form $k[N]$ where  $k$ is a field and $N$ is a set of nilpotent elements. So in fact, in Qiaochu's example it is enough to take the simpler quotient $k[x_1,x_2,...]/(x_1^2,x_2^2,...)$.
Also note as in the examples of the form $k[N]$ given above, in any example the maximal ideal will need to be be infinitely generated since a ring is noetherian iff every prime is finitely generated.
I do wonder if there are examples not of the form $k[N]$.