The numbering system for just about all Chicago roads ends at the Chicago city limits, except for some of the streets on the south side, which continue all the way down into Will County.
Even stranger is that there is a pocket of streets in Dyer Indiana that are numbered according to the Chicago system, as if they had expected the street grid to expand that far south and east. Expand to Hammond, Schererville, St John and you'll see east/west streets that are numbered in the 40s (the Gary Indiana scheme), then in the high 90s (Lake County Indiana scheme?), then the 210s (Chicago scheme), then the 70s (back to the Gary scheme).
EDIT - Chicago did a mass street renumbering (with a few street name changes too) in the early 20th century. It would be interesting to know if some of the suburban street numbering schemes for the roads that cross municipal boundaries are still using the old system and that is why the street numbers seem to reset once you exit the city.
I was wondering about that, particularly for the named roads. I believe Pulaski/Crawford was always Crawford first until Chicago renamed their portion to Pulaski and Skokie kept the Crawford name. I recall my dad telling me that when he was a kid in the 50s Pulaski was Crawford all the way down. I think that might have been the case with Western/Asbury as well, but I’m not too sure on that one. I’m sure there are many other examples.
And here’s an unrelated yet interesting Chicago street fact for anyone still reading: Elston starts and ends at Milwaukee Ave., so there are two Elston/Milwaukee intersections.
Unincorporated addresses in Kane and DuPage Counties do use a reference system based off State & Madison as baseline (although it's been codified based off county and township lines). Eg. 40W100 Keslinger Rd is a bit over 40 miles west of State St.
They have their own section in the postal addressing standards, under the "Unusual Addressing Situations":
I spent 20+ years in Washington, DC so am used to NW, NE, SW, and SE with avenues, numbered streets, flower streets of two syllables, blah blah blah. But I now live in rural Virginia and state routes are the norm.
Your explanation got me to focus more on what and why. And, I learned something, so thank you.