Most of my photographing is outdoors while hiking, so I have some experience with this.
There is no one answer, since it depends on how much you are willing to take (lug around) versus how much flexibility you want in capturing something you see. The longer the hike, the more you may want to minimize the gear and live with the realization that you're just not going to capture everything as you would if you had your full kit available.
The amount of time you have for the distance also matters a lot. Is this a 10 mile hike you only have one afternoon for? If so, keep yourself unencumbered. If you've got all day and easy abort paths back to your car, then you can take a lot more, bump around slowly, and quit if you've had enough.
That said, my minimum gear is my camera (Nikon D3s), 24-105mm zoom lens, and polarizing filter. This is fine for just about all landscape photography applications. The polarizer may sound like a frill, but it's useful for the majority of landscape shots. Unless it's totally overcast (not good for lanscapes anyway), you can always make the sky look a little nicer. Even if the sky doesn't matter or isn't in the picture, the polarizer will be useful for deciding whether you want the sheen off of leaves and puddles and the like, or want to reduce it. Both can be useful, depending on the scene and what you are trying to show. Particularly for landscape photography, don't leave home without a polarizer.
If you are into wildlife, the kit above won't help much, but then again that's a much harder problem. Serious wildlife photography requires a long lens (too much is never enough), a good tripod, and lots of time. Casual wildlife photography requires a long lens, hope you'll find something to brace yourself against where and when you need it, and lots of time. A polarizer is much less useful. I usually don't bother putting one on a long lens. Besides, with a long lens every bit of light is useful.
I disagree with others in that I recommend not trying to bring a tripod, especially since you said your main interest is landscapes. Think about it. Landscapes are by definition in the open, which usually means plenty of light. Generally they are taken with wide lenses. The problem is usually getting the whole scene into the picture, not the details of something small and far. The combination of plenty of light and wide lenses means that hand-holding is really not a problem. Tripods are heavy, but most importantly are unweildy and take time and trouble to set up. You just don't need one, and the disadvantage of having to lug it around, probably requiring a larger pack and system for strapping it, outweigh the advantage of being light and more portable and less of a chore to set up in most cases. If you want to do velvet flowing water with a multi-second exposure and ND filters, then you need a tripod. Leave the tripod and ND filters back in the car unless you know that's what you're going after and are dedicating the time and effort to it.
Beyond the 24-105mm zoom, then next thing I take, which is most of the time, is a 60mm macro lens. That's a personal thing whether you're into flowers, bugs, moss, and the like. Personally I find the small world both fascinating and easily overlooked, which leaves a lot of possibilities for interesting pictures. If you're not into that, forget about the macro lens.
When I want to leave more options open, the next thing I take is a 300mm lens. Often I don't use it at all, even on a all day hike. There are only limited things you can do with it, but if you encounter the right situation there is no substitute for a long lens.
Another area I disagree with others on is worrying about weight too much. Again, think about this logically. If you're on a short hike of only a few miles or less, then a extra few pounds aren't going to make any real difference. Your day pack will feel about the same whether you throw in a extra lens or two or not, even if one of them is a 300mm. The real issue is space and klunkiness, not weight. On this type of hike I keep the camera with one lens on it (usually the 24-105mm) around my neck and over one shoulder while walking. That allows it to be grabbed quickly, but it doesn't dangle like it would just around the neck. That's most of the photo-related weight right there, and it's not even in the pack. There should be plenty of room for another lens or two in the pack.
On long hikes, again a extra lens or two won't matter much because its weight will be small compared to the water, wind breaker, flashlight, etc, you are going to bring anyway. That extra 60mm macro lens if pretty much free either way. The 300mm lens is more of a size problem than a weight problem.
All this of course assumes you have a good and comfortable day pack. Get the kind with a layer of foam sewn in up against your back. Camera gear is rigid with hard edges, and can easily dig into your back if not positioned right. The foam padding makes that situation a lot better.
