From: 'Littledean', A History of the County of Gloucester: Volume 5: Bledisloe Hundred, St. Briavels Hundred, The Forest of Dean (1996), pp. 159-173.
URL: http://www.british-history.ac.uk/report.aspx?compid=23258 Date accessed: 15 November 2013.

and

"and in the 18th century Littledean bailiwick, which had become known as Badcocks Bailey, ceased to be separately distinguished."

From: 'Forest of Dean: Forest administration', A History of the County of Gloucester: Volume 5: Bledisloe Hundred, St. Briavels Hundred, The Forest of Dean (1996), pp. 354-377.
URL: http://www.british-history.ac.uk/report.aspx?compid=23268&strquery=bailey
Date accessed: 15 November 2013.

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Roughly, since the 12-1300s each Royal Forest was governed by a Constable; in our case from St Briavel's Castle who had overall control over ten senior foresters (bailiffs) each administering an area called a bailiwick, hence bailey.

Who Badcock was I'm not sure yet...perhaps the Littledean bailey Forester ?
There are just 41 Badcock PRs in this site's database, none obviously in Littledean that I've found although there are a couple in nearby Abenhall & Flaxley in the 1600s.

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UPDATE: Thanks muchly Admin for finding the earlier thread on this subject, I DO wish I could also remember to search for related old threads BEFORE trying to answer queries, duh...
:-)

Broad Street
Little Dean

Caroline BRAIN - 86 Widow - born Little Dean
Sarah Eliza WESTBROOK, Niece - single - 40 - born Little Dean

Can only see the above on the 1911 census Broad Street, and Caroline seems to have lived there in previous census returns.

Ancestry Probate

BRAIN Caroline of Littledean Gloucestershire, widow died 28 March 1920.

Probate Gloucester 18 May to Sarah Eliza Caroline WESTBROOK, Spinster, and Ralph Edwin ALLEN retired Civll Servant.

Effects: £4,722 17s 3d.

Caroline was married to Samuel BRAIN

Ancestry Probate

BRAIN Samuel of Littledean Gloucestershire, farmer died 15 January 1892 Probate Gloucester 7 September to Caroline BRAIN, widow and Albert Esau BRAIN Farmer. Effects £143.

I know the area well but not under that name

me too, it had me stumped for a while, I was looking at an old Collafield map and it caught my eye !

so looks like a transcription error to note

http://www.old-maps.co.uk/
co-ords 367310,214508

THE LONDON GAZETTE, JULY 28, 1863.

TO be sold by auction, pursuant to an order of the High
JL Court of Chancery, made in the approbation of the
Vice Chancellor Sir John Stuart, and with the concurrence
of the mortgagees, in one lot, by Mr. James Kura, the
person appointed by the said .Judge, at the King's Head Inn,
in Littledean, in the county of Gloucester, on Monday, the
31st day of August, 1863, at six o'clock in the afternoon :
A freehold dwelling house with the outbuildings and
piece of garden adjoining, situate at Badocks Bailey, near
the turnpike road leading from Littledean to Guns Mills, in
the township of East Dean, and county of Gloucester,
occupied from year to year at the yearly rental of £12 by
William Goold.

The premises may be viewed by permission of the tenant,
and particulars and conditions of sold may be had (gratis)
of Messrs. Poole and Johnson, Solicitors, No. 9 and 10,
New-square, Lincoln's-inn, London; of Mr. James Knight
Smith, Solicitor, Newnham, Gloucestershire; of the
Auctioneer at Gloucester; and at the King's Head Inn,
Littledean.

do you have any further context to help - is it a census transcripion or record for example ?

Bailey Hill Bailey Inn Yorkley etc