"That's "Clunker No. 1", an Epiphone that Les modified so heavily and frequently that he had to put a steel plate on the top to hold the guitar together. Note the bar bridge"
Also know as the log - actually this was the guitar that Les brought to Gibson as his prototype for a solid body jazz guitar and later becaume the Les Paul Model... but as said it originally was an Epiphone not a Gibson. That's why it has no name on the headstock
