groze: I assume the DRM-free GOG release of the game lacks any daily stuff, as the only thing I see in the menu is Play and Co-op, other than the usual Settings, Stats and Quit. When you enter play, all you get is a character select screen, nothing more.

I tried taking screenshots of the game, so you could check it out for yourself, but it seems I can't, for some reason (Print Screen doesn't seem to work with Nuclear Throne). Still, for what it's worth, I'm having a blast with the game either way, even without daily challenges and missions and whatnot. Maybe they'll implement that at a later date? As far as I'm aware, the game was only officially released a month ago, perhaps they are still building things up?

Still, answering your question directly: no, as of now, there aren't any daily missions available in this GOG version.

Zoidberg: Are they implemented in this drm free release or have they just removed the feature like Spelunky's devs did?

tremere110: They aren't in the Steam version either:
link

Zoidberg: Mmh, interesting. But it's temporary.

But the fact that they use a third party server before the upload to the steam servers could make it possible to have that feature on drm free versions.

Any devs here to answer this question?

pe7ouill3: http://steamcommunity.com/games/nuclearthrone/announcements/detail/870689050348007756
The answer is here (fixes) : They remove the runs for DRM free version in the update 98.

Zoidberg: Wow, this sucks... if I understand correctly... have they removed them or reintroduced them in this update?

pe7ouill3: Removed. It was already done on GOG release but it's for the DRM free version sell on humble store. But that confirm that they don't want to implement runs in DRM free version (GOG and humble)