What counties can you name, that don’t have the name of the city in them? (And obviously the ones near to where you’re from are less interesting for the purposes of this question, so I’m leaving out the ones from the Bay Area and Boston)
Here’s the ones I can think of:
Kings
Queens
Dade
Cook
Cuyahoga
Orange
Dade and Broward 
Get me out of there and I’m “uh, what’s a county?”
Harlan County, U.S.A.
Hazzard County
Brisco County, Jr.
I could recite quite a few from the California map that used to be on my wall.
But what do you mean by “the city”? Santa Clara is in Santa Clara Co., but the area is pretty much dominated by the county seat of San Jose.
Ok, I guess that’s a grey area. I’m thinking more of counties like San Francisco, Los Angeles or Dallas.
There are also oddities like Baltimore City, which is an entity unto itself, and not in Baltimore County.
I think this question is mostly “who spends a lot of time looking at maps and/or takes a lot road trips”.
If presented with state maps with only county borders outlined(no names) I could place:
Maryland - all 23 counties plus Baltimore City
Texas - about 75 to 100 counties
Oklahoma - maybe a dozen
Louisiana - 10 parishes I can think of right now
Virginia - about 20 counties and independent cities(a la Baltimore)
Delaware - all three counties
Plus somewhere between three and ten counties in each of PA, FL, AR, NM, NV, MT, WV, WI, NY, NC, CA
OK, outside of my state (OH) and places where I’ve personally lived:
All of the boroughs of New York, though I know that some of those have different names than the boroughs, so I might get some of the names wrong.
Arlington, VA (DC light)
Orange, CA (approximately Los Angeles)
Maricopa, AZ (where Arpaio was sheriff of)
Cambria, PA (because my grandparents lived there)
I think there are a few others I know, but they all share a name with their principal city.
In the Chicagoland area:
Cook
Lake
DuPage
McHenry
Kane
Will
In Illinois:
Boone
Ogle
Jo Daviess
Kendall
Grundy
Mason and Dixon
Harris, Rusk, Brazos, Galveston
Oklahoma, Logan, Canadian
Larimer
Dade
Cook County. I have The Good Wife to thank for that.
Since newscasters never mention a New Jersey municipality without giving the county it’s in, all 21 of them.
All 83 counties of Michigan, of course.
Cook and Sangamon Counties, IL
Alameda Co, CA (Because of the stadium name).
Most of the ones in PA including all of those around Allegheny County and Luzerne County. I also know a few in Ohio (like Shelby County) and Indiana (such as Tipton County). But in the interest of full disclosure its because I love history and also tend to enjoy county fairs.
At the end of The Blues Brothers, Jake and Elwood are looking for the office of the Assessor of Cook County.
I’ve also heard of Marin County in California; I believe it’s just across the Golden Gate Bridge from San Francisco.
In Oregon:
Multnomah
Washington
Deschutes
Malheur
Jefferson (the town of Jefferson is actually in Marion County)
Morrow
In Alaska, it’s boroughs. There is no town of Bristol Bay in Bristol Bay Bourough.
Also:
Northwest Arctic
North Slope
Aleutians (east and west)
Lake and Peninsula
I can name several dozen in my native Illinois, likely two dozen here in Missouri, another dozen in Florida. Does the OP really want me to list them all?
In the states where I don’t know a dozen or more:
Orange, California
Stanislaus, California
Tuolumne, California
Mendocino, California
Marin, California
Barnstable, Massachusetts
Orleans (Parish), Louisiana
St. Tammany (Parish), Louisiana
boroughs : Brooklyn/Bronx/Manhattan/Queens/Staten Island…
counties : Kings/Bronx/New York/Queens/Richmond
Not the Bronx, it’s borough/county/“city name”, because Bronx, NY is what you’d put on an envelope to someone living there. Queens works because all addresses in use their town/area, like Flushing. Forest Hills, Long Island City… There’s no “Queens”.
Bunches, in part since my work-study job in college required me to do data entry on addresses at the county level. 
Off the top of my head (and, as per the OP, not including counties that share a name with a major city in the county):
Wisconsin:
Brown
Outagamie
Calumet
Dane
Marathon
Oneida
Ozaukee
Trempeauleau
Door
Illinois:
Cook
DuPage
Kane
Grundy
Lake
Winnebago
Ogle
Sangamon
Will
Minnesota:
Kandiyohi (I just love that name!)
Ohio:
Hamilton
Cuyahoga
Michigan:
Wayne
There is a surprising number of Douglas Counties in the various 50 states, but there is no “Douglas” in Douglas County, Nevada.
Fulsom
Calaveras