www.alphadictionary.com

Mense, as in to have one's meat and mense - when you invite someone to dinner whom you know can't come. You have shown generosity (mense) and it has cost you nothing (you still have your meat).

I will have to start using this word; it is certainly a good one. I wonder if there is any connection with Immense

To answer the last question, I believe not. Dictionary.com says mense is akin to mensch (GW Word here & here). Immense is from not mensus, not measurable. Etymonline doesn't have mense, but it does have the same letters ending in s, menses. Which is pretty much what I see when I see this word.