responsive website min-width?

In my experience most responsive websites don't have min-widths so when resized to a tiny size (say a 100px window) the content screws itself up and becomes illegible.

example: http://alistapart.com/article/responsive-web-design

I love the flexibility of RWD but usually implement a min-width on my sites as I don't like the site presentation when it gets too small. A lot of sites which aren't responsive let you scroll and I was wondering if it would be a good compromise to have a min-width and whether it effects usability at all. examples https://theartofdining.co.uk/