Brown Drip?

If the chimney and cap are galvanized steel, it could be rust.

It's so far from the chimney though. You obviously have a roof leak where the valley meets the soffit. I'm not sure it has anything to do with your chimney.

Agree with webby, you have a roof leak. Where that is coming down the wall under the soffit, there is a problem up there somewhere and there could be more water behind the siding that you don't see. The color could very easily be from a little soot on the roof picked up by the rain.

If it makes you feel any better I have the same chimney and the same problem. I think it is a by product of the burning process as well and since my roof is covered with snow and ice I suspect it just runs down the chimney and down the siding. Good to know it comes off easily.

everything is clean in the attic?

You have a roof leak, likely caused by improper installation of the chimney/flashing, possibly unrelated and just a leak for other reasons. What you are seeing is likely tars from the roofing.

If your gutters are draining properly and not full of leaves, then you have a roof leak. That is a good foot + back from the drip edge of the roof. No way for water to get back that far unless something is backing it up. Make sure where the two roofs combine there isn't a bunch of moss or anything as well.

You need to look in the attic at the underside of the roof sheathing. Stains, discoloration, mildew, etc. would all be signs of a roof leak.
The stain has that tea-like color that occurs after water passes through old oxidized and dirty attic framing.

In our case we have a cathedral ceiling so I can assure you there is not leak, we would see it in the house. I suspect that this is the by product of burning and perhaps it occurs when it is either snowing or raining. The chimney itself is stained but it eventually washes itself clean. I cannot see any other reason for it. The chimney was professionally installed and inspected.

I would pull the soffit and have a look

Offset said:

In our case we have a cathedral ceiling so I can assure you there is not leak, we would see it in the house. I suspect that this is the by product of burning and perhaps it occurs when it is either snowing or raining. The chimney itself is stained but it eventually washes itself clean. I cannot see any other reason for it. The chimney was professionally installed and inspected.

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Please reconsider your statement of "so I can assure you there is not leak". you have posted a picture of a leak, the only question is where it is coming in at, not if it exists. You can ignore it and over time the leak will cause rot, eventually you will see it at other places too. It is common for water leaks to show up many, many feet away from where they start in at.
As someone who has repaired the bad workmanship of others all of my life, please let me assure you that the just because you paid a professional to do the work does not mean it was done properly. Also, inspectors are often no better than the person doing the work. Some are good (or even great), some are careless, (or even negligent). Without inspecting the roof there is no way to really know for sure. It would have been very easy for the contractor to accidently poke a hole in the shingles or even crack some if it was cool when they were up there. Also your valley has the second course of shingles (the ones on your main roof, not the dormer) unusually low. They are generally not supposed to be ran to the bottom of the valley. This could have been done to hide an under wrap from the shingles on the dormer onto the main roof. You are also pretty far north, not sure if you have ice damming in your area, but that can easily make water run up and under the lower shingles too.

Even if you don't have a leak, you can also get condensation under the roof deck and that will turn brown and run down the inside of the soffit.

sportbikerider78 said:

Even if you don't have a leak, you can also get condensation under the roof deck and that will turn brown and run down the inside of the soffit.

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Very true. Generally in houses with condensation issues building up on the underside of the roof it will happen on more than one area. I have pulled down drywall/insulation in the winter time and seen 2" of ice stuck to the underside of the roof (not here in Oregon, I am from Alaska). It is from too high of humidity/poor vapor barrier/poor insulation/poor venting and more. The fact that it is showing up at the point of the valley indicates a leak, it would be in a more spread out manor if it was condensation.