What counties have you lived in? (You can define “taken up residence in” however you like, but I’m considering any county Ithat 've registered to vote in during my life).
Me:
Wayne County, MI
Cook County, IL
Kent County, MI
Ottawa County, MI (current)
Santa Clara, CA (for only three months, but it’s where I was born)
DuPage, IL
Brown, WI
Dane, WI
Cook, IL (since 1990)
I think I may be kind of boring in this respect. 
Bronx, NY
Tompkins, NY
Lane, OR
Boulder, CO
Washington, DC (no counties)
Wellington Region (county equivalent, I guess), New Zealand
Auckland Region, New Zealand
Canal Zone, Panama (no counties)
Panama District, Panama Province, Panama
Polk County, IA
Allen County, OH
Mason County, WV
Marshall County, WV
Ohio County, WV
Monongalia County, WV
Anne Arundel County, MD
Baltimore County, MD
York County, PA
We don’t do counties in Aus, but I’ve lived in four states: born in and most of my life in Victoria, but also in New South Wales (right on the border with Vic), far north Western Australia, and now in Far North Queensland.
All in New York (although I’ve spent quite a bit of time in Texas): Chemung, Clinton, Dutchess, Erie, Genesee, Livingston, Monroe, Orange, Putnam, St Lawrence, and Ulster.
These are all I remember:
King (WA)
Maricopa (AZ)
Pulaski (AR)
Lincoln Parish (LA)
Ouachita Parish (LA)
Aberdeen/Peterhead (Scotland - don’t remember if they have counties)
Rotterdam (Netherlands - ditto)
Ventura (CA)
Parker (TX)
Tarrant (TX)
Bucks County, PA - until I was 3
Montgomery County, VA - during college
Burlington County, NJ - the whole rest of my life
Washington County, Wisconsin
Waukesha County, Wisconsin
Milwaukee county, Wisconsin
I own 2 cabins, don’t know if you include that as a residence:
One in Barron County, Wisconsin
The other in Dodge County, Wisconsin
I own a small place in Sarasota, Florida as well.
St. Louis County, Missouri
Monongalia County, West Virginia
Macomb County, Michigan
Oakland County, Michigan
Cook County, Illinois
Lake County, Indiana
I have also lived entirely outside of any county in the US, while still residing in the US. One city I’ve lived in is an “independent municipality” and is not within any county whatsoever.
Shelby County, Tennessee
Henry County, Iowa
Ringgold County, Iowa
Jefferson County, Iowa,
Buchanan County, Missouri
Phelps County, Missouri
Benton County, Arkansas
Douglas County, Nebraska
Segdwick County, Kansas
Harris County, Texas
Surrey, England
Berkshire, England
Hampshire, Engand
Ingham County, Michigan
Eaton County, Michigan
Lake County, Ilinois
Polk County, Iowa
San Diego County, California
New London County, Connecticut
Hampton, Virginia (independent city)
Newport News, Virginia (independent city)
Norfolk, Virginia (independent city)
Honolulu County, Hawaii
Argyll and Bute (council area), Scotland
Broward
Dade
Palm Beach
Nassau
Cook
Kalamazoo
Kent
Places where I was actually a resident:
Baltimore county, MD
Anne Arundel county, MD
St. Mary’s county, MD
King George county, VA
Duval county, FL
Clay county, FL
Places I lived when in the Navy:
San Diego county, CA
Solano county, CA
Tippecanoe county, IN
Shelby county, TN
Orange county, FL
Escambia county, FL
Newport county, RI
Clarke county, GA
Not very many for me.
Chautauqua and Tompkins, NY
Sarasota, Orange, and Brevard, FL
Oddly enough if you add the time I’ve spent in Alexandria VA, and Ontario and Saratoga counties in NY, it’s probably around half a year but it was in 3 and 4-day spurts.
Monongalia (WV)
Tarrant (TX)
Harris (TX)
Los Angeles (CA)
Johnson (KS)
Hinds (MS)
St. Joseph (IN)
Marion (IN)
Douglas (NE)
From birth to present:
Los Angeles, CA
Santa Clara, CA
Alameda, CA
Clark, WA
King, WA
Washington, OR
Los Angeles County, CA (where I was born);
Orange County, CA
Hiyama Subprefecture, JP
San Diego County, CA
Los Angeles, County, CA (again)
Whatcom County, WA
New York, NY - the county so nice they named it twice. Of course, the only U.S. elections I’ve ever voted in were in 1992 and 2000.