EA will soon be adding Star Wars: Battlefront to their list of timed demos available through Origin. You can see the Star Wars themed title in the catalogue of timed Origin trials, complete with “coming soon” message.
Looks like you’ll get four hours free with this one, which is probably enough time to figure out whether you want to carry on playing Star Wars: Battlefront or not. Existing timed demos include Plants vs Zombies: Garden Warfare 2 (you get 10 hours of that one), and the first level of Unravel. The Sims 4 and Titanfall have 48 hour demos available too.
The time-line for when Battlefront will be available is a bit vague, but if you keep an eye out it’s bound to turn up in the next few days.