I’ve flown non-transponder equipped airplanes, so I will elaborate on this. I am also going to paraphrase the regs and hope I don’t leave anything out or remember the regs incorrectly, because I don’t think Mangeorge really wants chapter and verse on this and my aviation stuff is packed away at the moment.
Normally, if you don’t have a transponder (and for all practical purposes these days we’re talking about a Mode C transponder, with location and altitude reporting, although there are still a few older models out there still flying) you can’t enter airspace that requires one.
Normally.
There are a couple of such airplanes that have been in the Mode C airspace around O’Hare since before that airspace existed and they have been granted permission to transit between their home base airport (non-towered, on the border of the Mode C) and airspace outside the Mode C, but only for that purpose - they can’t go hang around turning circles over the Loop.
ANY aircraft without a transponder CAN request a waiver to fly into any airspace. You just have to ask in advance. In some cases, this can be as simple as calling an airport tower, explaining you have a non-transponder plane you’d like to fly into their airspace/to their airport (you also do this when you don’t have a radio and want to enter airspace that normally requires a radio). The tower will either say “yea” or “nay”. In other cases you have to ask well in advance, in writing, and contact both the airport tower and/or the local Flight Standards District Office. In some cases permission is quickly granted. In others much more information may be required.
As an example, I know of a group of ultralight flyers who wanted to attend a fly in at Meigs Field in downtown Chicago (obviously, this was some time ago since Meigs no longer exists). Not only did none of them possess a transponder, most of them had no radios either. Ultralights are so small and slow air traffic radar doesn’t always track them very well (ATC systems edit out certain things, like birds, based on size and speed - a very small, very slow aircraft may be mistaken for a bird by the system unless it has a transponder). Obviously, this is a concern when they are requesting permission to transit some very, very busy airspace. They were required to detail very specifically their proposed flight plan, including planning for emergency landings, a very specific time window in which to fly, at least one of the two pilots who had private licenses were required to lead the group and carry a hand-held aviation transceiver (about the size of a walkie-talkie - I own one myself, as do a lot of other pilots), calling the controlling air traffic control agency via cellphone just prior to take off and just after landing, and so on and so forth. And they were granted permission and did succsessfully fly in and out.
Quite a few of the antiques you see in airshows, such as Stearman biplanes, may be without radio or transponders and may need to frequently request such permission to participate in airshows such as Chicago’s annual Air and Water Show.
Damaged airplanes may also request an exception to the rules as part of a ferry permit when being transported to a place for repairs.
In other words, there is a mechanism for waiving the requirements, but you have to have a reason. Depending on where you’re going and what you’re doing it can range from “I want to attend the pancake breakfast sponsored by your local flying club” to “I am being paid a gazillion dollars to perform outside loops at an airshow.” Different locations will have different requirements based on traffic, safety, and yes, sometimes local politics (sigh). Reasons for flying into O’Hare International without a radio or transponder will have to be a LOT more creative than requesting permission to fly into, say, Gary airport in Indiana.