What cameras (DLSR & lenses, Mirrorless or P&S) can be used when its raining cats and dogs without any kind of special rainproof accessories to protect them?
There are plenty such models with some in every category.
The highest resistance to inclement weather are waterproof cameras which are all point-and-shoot model with a sealed non-extending lenses. These can even go underwater between 3 to 12m for a minimum of 60 minutes.
Weather-poof cameras are sealed against splashes in any direction but are not protected against water pressure. For an interchangeable lens camera, a weather-sealed lens must be used to make the whole thing weather-sealed. In some limited cases you need to add a filter to the lens to complete the seal.
Weather-proof lenses are available for various mounts but some like Sony Alpha and Micro Four-Thirds mounts have an extremely limited number of weather-proof lenses. There are none for the Fuji X mount. Olympus makes a weather-sealed Four-Thirds to Micro Four-Thirds adapter so you can use weather-sealed Four-Thirds lenses on Micro Four-Thirds cameras.
There are several cameras and a handfull of DSLRs - all from Pentax - which are also freezeproof which makes them suitable for extended usage at sub-zero temperatures. Living in Canada and having owned all such DSLRs, I can attest to this. I have also taken my K-7 and K-5 in heavy rains and in several desert sand-storms and have had no trouble rinsing them (with lens attached) under the tap to clean them after. Needless to say, I am not worried about rain :) Incidentally, Pentax also makes the cheapest weather-sealed DSLRs and cheapest weather-sealed lenses.
I can vouch for the Canon 1 series, I've used both the 1Ds mkII and 1D mkIV for extended periods in the driving rain. You need a weather sealed lens, most L series are, you can usually tell by if there's a rubber skirt around the mount. I don't bother with a filter as the lens hoods on the primes I shoot with a really deep and stop water going anywhere near the front element (just don't point the camera up). Other than that you're good to go, these bodies can cope with pretty much everything except submersion.
I should note that other manufacturers offer weather sealed cameras/lenses too, but I can't personally vouch for them.
Professional level DSLR cameras many times can be used without issues in the rain. Some of them require a filter to complete the weather sealing of the lens. There isn't necessarily a "kind", just look for "weather sealing" and to what degree it is weather sealed. Note, you would want a lens and a DSLR body that is weather sealed, not just one or the other.
For point and shoot cameras, I would recommend any waterproof camera such as a Nikon AW100 or similar.
The necessity for waterproofness when working in rain is lessened by something obvious but important: it's raining. The rain gets on the front element/filter of the lens and leaves streaks or rain drops in your photo. The solution? Dry the element and try to keep it dry. How do you do that? Easiest thing is to keep the camera dry by sliding it into your jacket or using a raincoat of some sort. In the end, a weatherproof/waterproof camera is important to letting you work in those conditions for the peace of mind it brings, but practically, the solution is still to care for the camera and give it a dry spot.