Do only humans spend a lot of time daydreaming (or having "stimulus-independent thoughts")?
It seems that it is still a matter of debate whether animals are capable of mind wandering. For instance, there are a lot of publications about foresight, a future directed instance of mind wandering. Much of it comes from one group, e.g., 


  Suddendorf T, Corballis MC. (2007) The evolution of foresight: What is
  mental time travel, and is it unique to humans? Behav Brain Sci. 
  30(3):299-313; discussion 313-51.
  
  Suddendorf T, Corballis MC. (2010) Behavioural evidence for mental
  time travel in nonhuman animals. Behav Brain Res. 215(2):292-8.


But, for a review see: 


  Cheke Lucy G., Clayton Nicola S.. Mental time travel in animals. WIREs
  Cogn Sci 2010, 1: 915-930.


Or for a quite current example, see: 


  Osvath M, Karvonen E. (2012;7) Spontaneous innovation for future
  deception in a male chimpanzee. PLoS One (5):e36782.