When I studied music theory in college (years ago), I remember my theory instructor demonstrating a use of the omnibus progression where it would go on forever, sounding continuous to the untrained ear (which made the whole class start laughing after like a minute of increasing intensity). I think it involved making some tricky inversions at some point, but I can't find an example of anyone doing this online. Does anyone know how to make this never-ending omnibus progression?
To respond to something that came up in comments: to be clear, the Omnibus Progression doesn't need to be "modified" to be played indefinitely. If you look at the Wikipedia page first example, there's only an octave given in the example, but you end up with a G7 chord that is precisely the same voicing as the E7 seen as the fourth chord in the sequence.
Basically, the progression starts over again at the 10th chord, just now transposed "up a third." Every fourth chord is a dominant seventh, and they change voicing, but the voicing repeats every tenth chord. (Effectively, there are three different upper voices and they each take turns doing various parts of the chord during each three-chord segment of the progression. After three iterations of that, they reset.)
Another way to think about it is if you look at the 10th chord in the Wikipedia example (on B-flat7) and now just pretend you were starting the progression over again transposed to B-flat, then you can keep going. After 10 more chords, you start again (now in D-flat), and continue indefinitely.
The oscillating pattern mentioned in another answer is also simple to make. The voice-leading in the omnibus is effectively reversible, as it depends on strong chromatic bass motion and contrary voice resolutions in the upper voices. So, you can use a chromatic ascending bass instead with similar upper voices, and the whole thing goes backwards. If you turn around after several chords (usually done at another dominant 7th), you can get an oscillation that sounds a bit carnival-like.
I've seen (but I cannot remember where) a cyclic type of omnibus. As the progress is just an extended voice exchange, one could exchange the voices back though that would give a <> style rather than a wedge. One cheat (which works for all progressions but may not be musically interesting) is to double everything at the octave (or make 3 or 4 octaves if possible) then transpose the low voices up in a staggered manner and transpose upper voices down similarly; the result will be a fake Shepard Tone.
Since the basic construction is a wedge (<) which has one-way movement, one must figure out how to make this repeat and sound natural. Either fade in new or both.
Further to the Wikipedia page already linked to by Athanasius in a comment, the omnibus progression can also be extended in such a way as to harmonise a descending chromatic scale in one part and an ascending chromatic scale in another, like this: