DIY 3D printing seems surrounded by Good Ideas that don’t happen, which led me to look up some of the early patents in the field. As nearly as I can tell, any bright idea one might have has already been patented; although you can usually get away with tinkering it up in your basement (because you’re not worth enough to interest the patent holder’s attorneys), anything beyond that will darken your skies with lawsuits.
The granddaddy of all 3D extrusion machines seems to be US5121329 (Crump → Stratasys 1992-06-09): Apparatus and method for creating three-dimensional objects
Exploring the patents referencing that one as a foundation should keep you busy for a while; the PDF has clicky links.
Some fine tuning on the theme:
US6085957 (Zinniel/Batchelder → Stratasys): Volumetric feed control for flexible filament
US5303141 (Batchelder/et al → IBM): Model generation system having closed-loop extrusion nozzle positioning
Congealing 3D objects in a vat of goo probably starts with 4575330 (Hull → MVP 1986-03-11): Apparatus for production of three-dimensional objects by stereolithography
Remember: I’m not a patent attorney and my opinion is worthless…