Games that normally need the Steam client to run will only run in offline mode for 10 days according to "Monokuma" in
***this post***But the truth is that if you take your laptop to your cabin by the lake to spend the summer where you don't have internet, it doesn't matter how much money you have paid on games, after 10 days nothing will start.
As Draclvr says it may very well depend on the games you have.
Here's a long reddit thread showing people have a variety of experiences with offline mode
https://www.reddit.com/r/Steam/comments/4k13km/i_will_not_have_internet_for_6_months_will_i_be/I don't know. I've been playing some casual games in offline mode for awhile now with no problems
Are you playing them on a computer without an Internet connection?