A later form of give someone an inch and they’ll take an ell, the phrase give someone an inch and they’ll take a mile means: the slightest concession will be unscrupulously exploited.

Note: However, the latter phrase has occasionally been used with positive connotations—cf., below, the quotation from The Canton Press (Canton, Missouri, USA) of 10^th December 1868, in which give them [i.e., your children] an inch and they will take a mile refers to the striving towards beauty.

The earliest occurrences that I have found of the phrase give someone an inch and they’ll take a mile are as follows, in chronological order:

1-: From an entry, dated Sunday 8^th October 1837, in the diary of the U.S. poet, essayist and philosopher Ralph Waldo Emerson (1803-1882)—as published in Journals of Ralph Waldo Emerson. With Annotations. Edited by Edward Waldo Emerson and Waldo Emerson Forbes. 1836—1838 (Boston and New York: Houghton Mifflin Company, 1910) [page 313]:

The young Southerner comes here a spoiled child, with graceful manners, excellent self-command, very good to be spoiled more, but good for nothing else,—a mere parader. He has conversed so much with rifles, horses and dogs that he has become himself a rifle, a horse and a dog, and in civil, educated company, where anything human is going forward, he is dumb and unhappy, like an Indian in a church. Treat them with great deference, as we often do, and they accept it all as their due without misgiving. Give them an inch, and they take a mile. They are mere bladders of conceit.

2-: From a correspondence from Lisbon, Portugal, dated Tuesday 14^th May 1844, published in the Evening Mail (London, England) of Monday 20^th May 1844 [No. 11,602, page 5, column 5]—reprinted from The Times (London, England):

The coolest and discreetest minds here are convinced that nothing in diplomatic relations with such politicians is to be effected by subserviency; […] that coaxing the Portuguese is the way to prevent concession; and, that if you give them an inch they will take a mile.

3-: From the Marysville Daily Appeal (Marysville, California, USA) of Tuesday 2^nd August 1864 [Vol. 10, No. 28, page 2, column 2]:

Never Satisfied.—The Copperheads are growling louder over the release of Bishop Kavanaugh than at his arrest. They are very hard to satisfy. In custody or at liberty, they growl and abuse Gen. McDowell. No credit is given the General for the act promptly relieving the Bishop as soon as he ascertained the charges against him were unfounded, but he is cursed the harder. This proves our position correct—that the better these traitors are used the more abusive and contumacious they will appear. Give them an inch and they will take a mile. Good treatment is not what they want. They wish to create a conflict, and fair treatment will not bring that about. A pretext for a disturbance will suit them better.

4-: From The Canton Press (Canton, Missouri, USA) of Thursday 10^th December 1868 [Vol. 7, No. 18, page 4, column 1]:

The Love of the Beautiful.—Place a young girl under the care of a kind-hearted, graceful woman, and she, unconsciously to herself, grows into a graceful lady. Place a boy in the establishment of a straight-forward, thorough-going business man, and the boy becomes a self-reliant, practical business man. Children are susceptible creatures, and circumstances, scenes, actions, always impress. As you influence them, not by arbitrary rules, nor by stern example alone, but in a thousand other ways that speak through beautiful forms, pretty pictures, so they will grow.—Teach your children, then to love the beautiful. Give them a corner in the garden for flowers, encourage them to put it in the shape of hanging baskets, allow them to have their favorite trees, learn them to wander in the prettiest woodlets, show them where they can best view the sunsets, rouse them in the morning, not with the stern “time to work,” but with the enthusiastic, “see the beautiful sunrise!” Buy for them beautiful pictures, and encourage them to decorate their rooms each in his or her own childish way. Give them an inch and they will take a mile. Allow them the privilege and they will make your home beautiful.

5-: From the Winnebago County Press (Neenah & Menasha, Wisconsin, USA) of Saturday 23^rd April 1870 [Vol. 7, No. 28, page 3, column 1]:

Give a liar but an inch to start on, and he will take a whole mile of lying invention; and then impudently assert that the mile is well supported by the foundation of an inch.

6-: From Brevities. Paragraphic Pencilings of Passing Events, published in The Pittsburgh Gazette (Pittsburgh, Pennsylvania, USA) of Thursday 29^th July 1875 [Vol. 89, No. 178, page 4, column 5]:

A young man from Steubenville came to Mr. Stoy, on Fourth avenue, about a week ago and placed in the hands of that gentleman a farm for sale, and has since been about the office waiting for a purchaser. Being allowed some liberties, the old saying “give a man an inch and he will take a mile” has been verified in his case, for he took the liberty to jump into a pair of Mr. Stoy’s best pantaloons a day or two ago, besides stepping into a pair of new boots, with which he walked off, and has not been heard of since. The sale of “that farm” seems to have been a put up job.

The phrase give someone an inch and they’ll take a mile soon became proverbial—as illustrated by the following, about “the Attorney-General-Marshal controversy”, published in The Daily Bulletin (Honolulu, Hawaii, USA) of Saturday 15^th October 1887 [Vol. 12, No. 1,765, page 2, column 2]:

The Marshal had made an attempt to be en rapport with the Attorney-General in regard to the dismissal of one man. But […] the Attorney-General then conceived the brilliant idea of encompassing a power which did not belong to him and he would reorganize the whole Police Department. The attempt to lengthen the inch, offered by the Marshal, into the proverbial mile, naturally enough caused the Marshal to decline to be thus overridden.