I understand that mirror lenses already have high maximum apertures and that often, the last thing one wants to do is let in less light, but is there a physical reason that one could not build a working mirror lens that did use a variable aperture? Is it impossible to put the aperture in the right place? Is there something about the optical pathway that would ruin the functionality of an aperture. The advantage would be that one could get greater depth of field.
There is no real theoretical reason a mirror lens can not have a variable aperture. In fact, many Newtonian reflector telescopes do have a primitive adjustable aperture for use when viewing very bright objects such as the moon.
This homemade one has three holes of different sizes. In normal usage two of the holes would be covered and light would only pass through the selected diameter hole to control the amount of light collected from bright astronomical objects. Such apertures only affect brightness, though. Depth of Field isn't really affected.
The lack of any catadioptric mirror lenses that are made with adjustable apertures are practical reasons.
Most mirror lenses made for photographic usage are catadioptric in their design. That means they combine some of the properties of a reflecting mirror with some of the properties of a refractor lens. Designers do this to create a lens with very long focal length in a form factor much more compact than a conventional refractive lens of the same optical power.
In most cases the front lens element is either a flat dust protector or a "corrector" plate that is either an aspherical refracting lens or a more simple meniscus refracting lens. For lenses with a front corrector plate, the light enters the front of the lens, is refracted by the front corrector plate, is reflected by the primary mirror at the back of the lens onto the secondary mirror located in the center near the front of the lens just behind the corrector plate and then reflected by the secondary mirror through a hole in the middle of the primary mirror at the rear of the lens. Lenses with only a flat front protector plate usually have correcting lens element(s) just behind the primary mirror that refracts the light passing through the hole after it is reflected by the secondary mirror. Lenses with a corrector plate may or may not have additional correcting lens elements behind the primary mirror.
Since the secondary mirror (which is sometimes only a silvered surface on the center of the rear side of the meniscus corrector plate in the classic Maksutov–Cassegrain design) requires an obstruction in the center of the lens, a conventional internal diaphragm iris is not possible with such a lens. An external aperture attached to the front of the lens, similar to the one for our Newtonian reflector above, could be used with a flat front plate design. But it would be unwieldy and increase the size of the lens. This defeats the whole purpose of a mirror type lens which is high focal power in a compact design.
Such an aperture would not affect DoF in any meaningful way, though. Just as with the Newtonian reflector above, the aperture would only affect brightness. DoF is defined by the F-number which is based on the mirror diameter in a catadioptric lens design. The brightness is defined by the T-stop, which is reduced by such an aperture, just as it is reduced by the permanent center obstruction.
There has been at least one mirror lens that did incorporate a manual aperture iris at the very front of the lens. It was available in many popular SLR mounts around in the 1970s and 1980s. In this position the aperture setting only affected the T-stop number of the lens, but not the depth of field. The Ohnar 300/5.6 Mirror (also sold under other names such as Hanimex, Makinon, Panagor, etc.):
Is there something about the optical pathway that would ruin the functionality of an aperture.
The aperture is necessarily a hole in the middle of the mirror. The larger the aperture, the less room there is for mirror surface without making the lens larger. If you don't mind carrying around a lens that's longer and wider, then perhaps you could afford to put a larger hole in the middle, but mirror lenses are already fat enough to be unwieldy.
The advantage would be that one could get greater depth of field.
I think you've got things backward: smaller apertures (higher f-numbers) give more depth of field. Larger apertures (smaller f-numbers) allow more light but give less depth of field.
In most catadioptric 'lenses,' the primary mirror is the aperture stop. It is quite a large surface, and because of the tighter tolerances around mirrors mechanically crowded. Large diaphragms are expensive and it would be difficult to implement a mechanism to control a diaphragm inserted on top of the primary mirror.
A makutsov design can be changed where the secondary mirror is the aperture stop, but that has worse coma, astigmatism, and distortion than the stop being at the primary mirror. A better performance at the full aperture is usually considered better than a worse design which you can close the aperture of.