The Full Monty
Complete, the whole thing.
The Phrase Finder is based in Sheffield, England so we have a special interest in this phrase since that's where the eponymous 1997 film was set. No definitive explanation of where the phrase originates has emerged though.
The most often-repeated derivation is from the tailoring business of Sir Montague Burton. A complete three-piece suit, i.e. one with a waistcoat, for a wedding etc, would be the Full Monty. There is plausible hearsay evidence from staff who worked in Burton's shops who confirm that customers were familiar with the term and often asked for 'the full monty' by name.
Burton opened his first shop in Chesterfield, Derbyshire, in 1904. The business flourished and he began opening a chain of shops in 1906. This included premises in Sheffield. He also opted to live in the city where, just 200 yards from where I now type this, he set up home in Violet Bank Road after his marriage in 1909.
Burton's name was certainly well known. The business went on to become world's largest wholesale bespoke tailoring service.
Another explanation of the phrase's origin is from a Spanish card game where the pile of cards on the table is called a 'monte'. This is plausible but I can find no documentary evidence to support it.
Yet another comes explanation from Field Marshall Montgomery's alleged habit of wearing his full set of medals, or his alleged insistence on his troops eating a full English breakfast every day. It's true that Montgomery was universally known as Monty, but that's where the circumstantial evidence for this derivation ends. The fact that there is more than one explanation that links Monty and 'the full monty' is somewhat unconvincing.
Although the phrase has been in circulation prior to the film there don't appear to be any instances of it appearing in print before the 1980s.
In the 1982 edition of Yellow Pages (UK's commercial phone directory) for Manchester North has an entry for a chip-shop:
"Full Monty Chippy The, 30 Townley St, Middleton"
The association of the phrase with the meaning we now understand is suggested in the shop's name. To get an unambiguous use of the phrase to mean 'complete, the whole thing', we have to wait until 1985, as here from K. Howarth in 'Sounds Gradely' (North West Sound Archive):
"Full-monty, everything included ... a thorough display - messing about."
The Montague Burton version seems the more convincing of the suggested derivations, but the evidence isn't sufficient to be sure.