Mass Effect, Dragon Age: Origins could possibly come to the service in the future.
Microsoft have started to make some of Xbox 360 titles backwards compatible to Xbox One, and the work progresses as the official launch of the service looms. One of these games is Mass Effect. Electronic Arts was asked on Twitter if those games could be part of EA Access as well, so subscribers get them for free like Dragon Age: Inquisition this month.
Electronic Arts answered, that they are "investigating adding backwards compatible EA games to EA Access in the future". So it's actually possible, that the paid subscription service will be extended to feature classic games as well.