Sure, I know that the Great Lakes technically flow to the Atlantic Ocean, but how navigable are those passages? Are those passages natural or a series of man-made canals and locks? What’s the deal?
They’re both natural and man-made locks. Whether an aircraft carrier fits thru all the locks, I don’t know.
The latter is the case; it appears that the Great Lakes’ route is made up of the Great Lakes Waterway and the Saint Lawrence Seaway (Wikipedia). The latter is too small for an American aircraft carrier, and at 740’x78’x28’ it is just barely too small to fit almost any of the foreign aircraft carriers listed in my copy of Jane’s; only Thailand’s Chakri Naruebet will make it at 599’x74’x20’. There are (according to Wiki) lake-only ships that are moving in the direction of supercarrier size, but I doubt that any of them have the draught of 37’ that appears to be the minimum for American aircraft carriers. Regardless, the proper answer is yes, an aircraft carrier can make it to Chicago, but only one such carrier is active right now. To get the full scoop, make friends with Wikipedia.
ETA: if6was9 is correct and I was initially wrong; there are both natural and man-made portions of any passage through.
ETA#2: being the first with a full answer gives me warm, fuzzy feelings. Which is good, because it probably won’t happen again around here. 
No modern U.S. Carrier could pass through the locks on the St. Lawrence Seaway, not even close. Here’s a Wiki page on the locks giving the max. vessel sizes:
As straight man says, Thailand’s HTMS Chakri Nareubet, which is 182.6m x 21.9m, with a draft of 6.2m, could in theory just make it up to Chicago (I’m assuming that the Chicagoans are paying for this – the Chakri Nareubet doesn’t spend much time at sea due to lack of operational funds).
However, there’s also a possible problem with getting under the Garden City Skyway bridge on the Welland Canal, which apparently limits the height of the ship (above water level) to 116.5 feet (35.5 m). Although I can’t find a good number for the height of the Chakri Nareubet, here is a picture; to my untrained eye, it looks more than 120 ft above water level. So, IMHO you’d have to do some surgery to the superstructure if you wanted to get it to Chicago.
I did wonder about height (Jane’s doesn’t include that particular statistic, unfortunately –nor did Wikipedia include it in the regular stats.) But seeing as the radar mast is much higher than the rest of the ship, one could reduce height quite a bit without too much effort.
Addendum to above post:
On reflection, I don’t think the Chakri Nareubet would make it, even if one did remove some of the superstructure to get under the bridges.
This site gives the beam dimension as 22.5m (78ft) (still just within the Seawaymax limits); however, there’s also a listing for “Extreme Dimensions”, with a beam value of 100ft (30.5m). Aircraft carriers tend to have pronounced “V” cross-sections, so I’d be concerned that the Chakri Nareubet would scrape the sides (above the waterline) of a lock when the latter is at low water. If you want to build a vessel with a beam close to the maximum permissible for a lock, you’d build it with a “U” cross-section rather than a “V”.
Regardless, after looking at some more pictures and remembering that the flight deck is nearly one hundred feet wide… it’s a whole lot higher than it is wide, forcing the conclusion that the ship is much, much too high.
So let’s agree to a reasonably certain “no.”
There goes my “first with a complete” answer schtick – it was complete, but also wrong! :smack:
Oh, let’s not be so negative. 
You could remove the carrier’s entire bridge and superstructure to bring the height down. I wonder how deep the draft is under the Garden City Skyway bridge? If you dredge that part, and really load the carrier with ballast, you might be able to squeeze it under. Then remove all non-essentials from the carrier inside and out, and let it float “high” through the locks, perhaps with tracked vehicles on the topsides of the lock to help keep it upright.
Before I start costing out large helium balloons, exactly how much are we prepared to spend on this? 
Ideally less than it would cost to build an aircraft carrier from scratch. 
Well, if you’re allowed to float the parts up to Chicago and assemble it in situ (which seems to go against the spirit of the thread title), and you wanted to keep the costs down, where better to look for an example than China?
[For those in a hurry, the most revealing photo on that page is here.]
Unfair – that wasn’t a ship at all!
Here’s a thought: the Ville de Bordeaux, a RO-RO that carries the A380 to construction. From Airbus:
Not sure about height (or draught, for that matter) here, either, but it’s a start. It carries aircraft, hence it’s an aircraft carrier, right? 
Alternatively, you could custom-build a replica of, say, the Hosho – it would definitely fit.
Or you could just get all Fitzcarraldo on its ass
2 US Aircraft Carriers used to operate on Lake Michigan: USS Sable & USS Wolverine.
Both were training Carriers, used in WW2 to qualify young pilots for Carrier takeoff/landings.
They were converted sidewheel vessels, steam powered, & the work was done in shipyards on the Great Lakes.
They were essential, because no regular carriers could be spared for training, & if training occurred in the Atlantic or Pacific, the traing carrier would be subject to Axis submarine attack. On the Great Lakes, this was not a problem.
Home port–Chicago, Illinois.
Postwar- both scrapped by 48.
Which, in a sense, is what was done in WWII - building carriers from scratch.
The Navy trained carrier pilots at Great Lakes Naval Base north of Chicago. For the purpose, two civilian ships had flight decks built on top of them and thus were aircraft carriers cruising Lake Michigan.
These were not, however, fighting ships but trainers, and apparently their flight decks were much lower to the water than on the fighting carriers at sea.
So, while there have been aircraft carriers in the waters off Chicago, they were built on the lakes, they didn’t travel to them via waterways leading to and from the ocean.
It is possible that no current carrier could make it up the Seaway. On the other hand, the Block Islamd, the Thetis Bay, and the Iwo Jima class of assault carriers (all built or converted in the 1950s) as well as the WWII era Independence class light carriers and all the WWII escort and jeep carriers could have made it easily. (I know, the Navy and Marines don’t like calling their assault ships “carriers,” but anthying that has a flat deck on top that can receive a C-2A landing qualifies as an aircraft carrier in my book. It is also probable that none of them are in service with the USN any more. I do not know if any were sold to other navies and remain in service.)
Sure it could reach Chicago.
Of course you’d need every human on Earth to help in the portage…
There was some talk about transporting the USS Wisconsin to Wisconsin (perhaps to the martime museum in Manitowoc), but unfortunately the locks on the St Lawrence are nowhere big enough (and I assume the same is true on the Mississippi)
Brian
What if the carrier were on a treadmill?
Water-mill?