tl;dr Yes there have been theorized, and extremely possible binary systems that have not been observed. One such thing is a TZO, or a Thorne–Żytkow_object. This is a neutron star-red giant binary.
There are many kinds of binary pairings that have not been discovered yet, if they exist. A hypothetical star is the quasi-star, which you could say it is a binary between a black hole and an extremely massive star. You can also consider a Wolf–Rayet star, which can occur from a binary system. Other binary systems would include theoretical stars, also known as exotic stars.
However, I do know of a binary system that is composed of real observed types of stars or stellar remnants. This system is composed of a red giant and a neutron star, with the neutron star spiraling in towards the red giant's core. It starts off with a binary system with a star and a neutron star. When the star becomes a red giant, its atmospheric friction on the neutron star(the red giant envelops the neutron star) makes their orbit deteriorate, and the neutron star and the red giant's core will spiral into each other. After the neutron star meets the core, if the mass of the core does not overcome neutron degeneracy pressure, then the core is replaced with a neutron star. If it does, then a supernova resulting in a black hole occurs. This binary system is particularly significant because this can shed light onto how fusion works, and how binary systems develop, because in this model the neutron star does not absorb a lot of the red giant's mass. The surface of the neutron star would also be so hot that fusion can occur, demonstrating a new way of fusion.
Astronomers can observe these stars by their emitted wavelengths because the fusion creates heavy elements that may find their way to the surface of the giant:
To look for a TZO, the team searched the spectrums for some of the elements predicted to be produced in high quantities in these objects: lithium, rubidium, strontium, vanadium, zirconium, and molybdenum. Not all of these are easy to image, so they focused on lithium, rubidium, and molybdenum. They compared the levels of these elements with elements that create spectral features nearby but aren't expected to be enhanced in TZOs: potassium, calcium, iron, and nickel.
I would say that no true examples of this kind of binary pairing has been found. However, there are three possible candidates that are observed, though recently, these stars are under serious questioning on their validity(that may not answer part of your question on "at least suspected :("), according to
There is another one mentioned by mistertribs(all credits to him), that a black hole and neutron star binary is not observed to exist. These type of systems are possibly the most interesting from a gravitational point of view. They should produce gravitational waves, which could be detectable by LIGO. LIGO has yet to observe such a system and confirm such, which seems a bit puzzling, although something like the blue supergiant in Cygnus X-3 could evolve into a neutron star in the future. This is a case where more data and observations are needed.
However, scientists were only able to constrain the merger rate enough to eliminate the most optimistic models, and that it would take several more observing runs with no results before a continued failure became problematic.