I feel your pain. I have been searching this too and I think it is way to complex to do this with Travis CI. It is possible, but it is a nightmare to setup. The process is (insufficiently) described here: Automating Jekyll/Github Pages builds based on time, justifying this question.
The reason that this is overly complex is that Travis CI is not meant for this job and is basically overkill. It would be really cool if Forestry.io or CloudCannon (or any other hosted Jekyll service) would implement this, so it would become just flipping a switch.
It is funny to see that this question of the OP follows this one: Count days in Jekyll. You get the impression that the OP thinks that daily builds are the solution for the 'counting days problem'. I think that the solution to that problem is Javascript. However, daily builds of a Jekyll site would enable other basic website functions, like 'post scheduling', which cannot be solved/is much harder to solve by Javascript.