I have some scrap notes, probably from a talk by Vezzani, where he compares a so called “ometric’‘ approach with the TV approach. Here Rings corresponds to Ab, Monoids with partially defined addition corr to the slice category of pointed sets over Ab (Kahn: But these are affine log schemes!), Monoids corr to Set, and Monoids with zero corr to pointed sets.
For blueprint geometry, see several papers by Lorscheid and/or Lopez Pena. Here is a blog post mentioning connections with Reineke’s observation, and explains that a blue scheme can be an elliptic curve: http://www.neverendingbooks.org/index.php/quiver-grassmannians-and-mathbbf_1-geometry.html
Takagi has an arxiv preprint on convexoid rings, claiming to construct a compactification of Spec Z.
http://mathoverflow.net/questions/1628/kf-1-sphere-spectrum
nLab page on Field with one element III