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?
If you allow for flat mirrors, I've seen five myself. Answer(s) to How did I flip some mirrors around in the dark at 3 AM and change the focal length of a 24 inch Boller and Chivens? show four mirrors and there was one more on the floor below to make the beam horizontal. If you count the aluminized reflective grating as a mirror it was six.
For more info, see Nasmyth and coudé focus
For more examples of three mirrors built into the telescope itself (as opposed to an eyepiece diagonal for viewing, see answer(s) to
For more little mirrors that send light to different instruments, see answers to
- What is a pupil slicer, and how does it work with anamorphic optics in VLT's ESPRESSO Echelle spectrograph?
- Where exactly is the modification that first corrected the spherical aberration in Hubble's primary mirror?
But three or more curved mirrors is interesting
One example is the Hubble Space Telescope using the Wide Field and Planetary Camera 2 or WFPC2.
Basically it is another whole telescope inside the Hubble, re-imaging the primary focal plane with a second purely reflective optical Cassegrain system to avoid chromatic aberration. (There would be no way to correct a fused silica lens system from 115 to 1050 nm.)
Figure 3.36 WFPC2 optics. Light enters the optical train from the main telescope at left.