At the start of "The Even More Incredible Machine" I'm prompted to put in a character code.
I've tried any code that comes to mind and it appears as if the game still works. However, this is a very disconcerting warning and intrudes upon my gameplay, therefore I will not mark this topic as solved until I get an answer from GOG staff that removes this screen all together.
Obviously, the game is still playable and it doesn't hinder game play as far as I can tell, but I would like no hint of DRM in my DRM free games.

This seems rather silly to me. It's designed to work no matter what you put in so it's not functionally DRM at all, and removing it entirely is almost certainly beyond what GoG has either the ability or rights to do.
I'm not crazy about the screen either, but it's in absolutely no way DRM at this stage.

I admit, it isn't necessarily the best request, but I will settle for a warning on the game page or a complete redesign of GOG's features.
GOG is in beta, so obviously I can't expect everything to work all of the time, but without pointing this (non)issue out, people won't know.
basically I made this post to enlighten those who are inclined to read the forums first - and hopefully prevent some nerdrage.

A warning of what? It doesn't feature any DRM, so they're living up to their end of the deal on that. Yes, the screen is still there, but since it's been deactivated it doesn't actually restrict anything.
I dunno, this seems like a purely idealogical beef rather than a legitimate complaint about something that's actually happened.

Quite a tempest in a teapot here.
I'm guessing that due to the way that bit of copy protection was coded it was far simpler to recode it to accept any three parts than it was to remove the screen entirely. And I'd hardly consider having to click a mouse button three times to be any kind of about-face on GOG's no-DRM policy.

Heh, I actually did the same thing the first 3 or 4 times. I guess playing the game got my creative juices flowing enough to think "Hmmm... I wonder if this would work..."

The Incredible Machine 3 will not allow me to sign in - thus I cannot save my progress. What do I do?

Try right click on the .exe/shortcut, properties > Compability tab and select compability with an older OS and see if it works