Isn't this really just similar to a freemium model? You get limited interaction forever and must upgrade to a paid plan if you get hooked and need more.
This would be a subset of Freemium. Freemium can be trialware (time based), crippleware (feature-based), metered (usage based, like Dropbox or Flickr), etc.
It'd be interested to see someone measure the effect of this. By offering a timed trial, you can (theoretically) create a sense of urgency. On day 7 of a 14 day trial, you can email "Only 7 days left! You haven't logged in much yet, so you better hurry up and take it for a spin!". If you removed that time pressure and replaced it with a "Looks like you haven't used OurApp much yet. You still have 14 logins left!", would it perform as well? From a marketing point of view, I think saying "take your time" wouldn't be a win.
It'd be interested to see someone measure the effect of this. By offering a timed trial, you can (theoretically) create a sense of urgency. On day 7 of a 14 day trial, you can email "Only 7 days left! You haven't logged in much yet, so you better hurry up and take it for a spin!". If you removed that time pressure and replaced it with a "Looks like you haven't used OurApp much yet. You still have 14 logins left!", would it perform as well? From a marketing point of view, I think saying "take your time" wouldn't be a win.