WBEZ's Curious City has the story:
Every town that folded into Chicago, from Lake View to Hyde Park, had its own system for naming and numbering streets. Some towns counted out addresses starting from the Chicago River, while others started from Lake Michigan. Some placed even numbers on the north side of the street, others put them on the south. Some even let developers choose their own street names or numbers if there wasn’t a lot of local opposition.
Edward Paul Brennan was a delivery boy for his father’s grocery store, and later a bill collector for the music company Lyon & Healy. He was so frustrated with the chaos of Chicago’s address system that in 1901 he came up with his own. But it would take him years to get it implemented.
Beginning in the 1890s he started a scrapbook, collecting newspaper articles about problems with city navigation or delays due to address confusion. Articles had headlines like “Streets in a Tangle. Visitors Lost.” One report tells about a doctor who couldn’t find a patient during a house call emergency.
Today, Chicago addresses increase by 100 per block, 800 per mile. (Miles are significant because of the way land is surveyed in Illinois.) It's an easily-understood system that makes it hard to get lost in the city.