This pdf file from Make magazine shows you how to make a three string cigar box guitar and matching amplifier. At an estimated cost of $15 to $30 this project uses no standard guitar parts: for example the machine heads are eye bolts and the frets are filed down nails held on with rubber bands. You could substitute some of the parts for real guitar components if you want to make a luxury version.
Cigar box guitars originated in the USA during the depression era, when poor people made their own instruments out of junk. They used an empty cigar box for a resonator. Many examples do not have frets and were played fretless or with a slide. Some have only 1 or 2 strings.