Do telescopes exist that reflect the incoming light more than three times along their length? Refractors only use the length of the telescope once, reflectors twice, catadioptric telescopes like those of the Schmidt-Cassegrain design three times. Have telescopes been built that reflect the incoming light at least once more from one end of the telescope to another?

Say, by adding a tertiary mirror which reflects the light forward again, towards a small quarternary mirror in front of the secondary, which ultimately directs the light though a small hole in the tertiary (and primary) mirror towards the eyepiece?
Note: The above description makes this question different from the similar-sounding Do triple or more times reflecting telescopes exist?
If your willing to accept more than 2 discontinuous mirrors, the Three Mirror Anastigmat has 4 passes along some/most of the overall tube length. An early working prototype example (which I've actually seen in person many years ago) was built at the University of Cambridge by Dr. Roderick Willstrop. The Institute of Astronomy has a page on the Three Mirror Telescope which includes the following optical diagram:

The 0.5 meter 3 Mirror Telescope (3MT) has a focal length of 0.8 meters, in an overall tube length of 1.2 meters producing a very compact telescope with a large field of view (5 degrees in diameter) with good image quality (&lt;0.33&quot; across the whole field of view). The quality of the site at Cambridge was not ideal for building on the progress of the prototype and a proposal for building a larger version at a better site was declined by the UK science funding agency of the time (SERC, which became PPARC and then STFC) in favor of buying into the Gemini collaboration of two 8 meter telescopes.
Some constructed or in construction examples of three mirror anastigmat include:

James Webb Space Telescope (see Figure 2 of this page on the JWST Telescope)
The Simonyi Survey Telescope of the Vera C. Rubin Observatory which will carry out the Legacy Survey of Space and Time (LSST) has M1 and M3 ground into the same piece of glass (see Figure 7 of the LSST overview paper) with a separate secondary M2 mirror and prime focus camera
The planned ESO Extremely Large Telescope will use 5 mirrors in a folded three mirror anastigmat design along with 2 flat mirrors which will have 5 bounces along all or part of the overall &quot;tube&quot; structure before exiting to a Nasmyth focus; see the not very clear diagram below:    from the ESO ELT optical diagram page, somewhat better diagram in  Figure 4 of Hippler 2018.