The first Chicago City Directory was published in 1839 when the city was just 2 years old. The earliest directory I have found on the Internet, however, is for 1843 published by Ellis and Fergus, Printers, in 1844 when the city was boasting a population of about 8,000 inhabitants. Many of Chicago's most famous citizens would not arrive until a few years later. Montgomery Ward was born in 1844; Potter Palmer opened his dry goods store in 1852; Marshall Field and George Pullman both entered the city in about 1856. Writer Henry Blake Fuller wouldn't be born in Chicago until 1857. But, the directory is interesting reading none the less and the early advertising by "subscribers" is occasionally quite good. Their similarity to today's phone books and yellow pages is immediately apparent.
Internet Archive has a copy of this early directory: General directory and business advertiser of the city of Chicago for the year 1844: with a historical sketch and statistics extending from 1837 to 1844. To be more accurate, it is a reprint published in 1903 and includes a brief history of the city. An earlier reprint (1896) can be found at All Ancestors and is much easier to read. It should also be noted that different reprints vary in content. For example: Judge Henry Fuller (grandfather of Henry Blake Fuller) was not found in the 1903 reprint but is listed in the 1896 version as "Fuller, [Judge] Henry, res Dearborn, bet Randolph and Washington [died June 22, 1879, aged 76."
Additional examples of complete Chicago directories found on the Internet are:
Chicago City Directory and Business Advertiser 1855-1856
Polk’s Directory for Chicago 1928-1929
For more information on Chicago residence and business directories, see
Guide to Jewish Genealogy in Chicagoland (lots of links here), The Newberry Library, and Cook County Ilinois Genealogy Trails.