Is it POSSIBLE to drive from Chicago to Seattle in three days?
Yes.
Mapquest tells me it can be done in thirty hours of driving -- thus, by driving ten hours a day with NO STOPS for scenic viewing, you could do it.

Is it WORTH IT?
No. You'd spend so much time driving that your trip would be little better than looking at some postcards. And the drop-off fee for the car rental would cancel any savings you might be planning.

Why not fly to Seattle, rent a car THERE, spend three days seeing the beauty in THAT area, return the car after a RELAXING three days, and then take the cruise?

My solution would be to fly to Chicago (early morning) and then go to lunch in Chicago before getting on the Empire Builder.
The Empire Builder takes 46-47 hours from CHI-SEA.
If you flew to MSP or MKE for a bargain flight, you could still ride the Empire Builder across North Dakota and Montana.
You don't want to pay a drop off fee on a rental car in Seattle just before you get on your cruise ship.
The Empire Builder leaves Chicago Union station at 2:15PM every day. If the weather is good, you can go to the top of the Willis (Sears) tower which is about 3 blocks from the station.

I've driven Chicago - Seattle on several occasions and would say that 3 days is too short a time to see much of anything besides I-90. I think exploring the Pacific NW for 3 days makes more sense and probably easier on the bodies and the pocketbook, but don't know if that appeals.

If you DID want to fly part of the way and drive the rest, then I'd probably opt for Sioux Falls or Rapid City instead of Chicago. South Dakota is marvelous for just a day or two - see Badlands NP (easy access off I-90) and Mount Rushmore, or visit Deadwood, then stop at the Little Big Horn battlefield in eastern Montana (also easy access from I-90) then roar across I-90 through western Montana and over the Continental Divide into the Idaho panhandle and across Washington State. It's a great drive with a tremendous amount of "old west" history and scenic diversity, but with three days you'd still be spending six or seven hours a day on the freeway.

Ok, so all of your responses have confirmed that I was a bit crazy to think this possible. Thanks for your input. I will look into the Empire Builder or flying into Sioux Falls.

The reason I suggested Minneapolis is because it is along the route of the Empire Builder. It leaves at 11:15PM and is only 37-38 hours from Seattle.
Amtrak does not serve South Dakota.
Fargo North Dakota would be your choice west of MSP.