Any reason to write the "private" keyword in C#?

AFAIK, private is the default everywhere in C# (meaning that if I don't write public, protected, internal, etc. it will be private by default). (please correct me if wrong).

This is not true. Types defined within a namespace (classes, structs, interfaces, etc) will be internal by default. Also, members within different types have different default accessibilities (such as public for interface members). For details, see Accessibility Levels on MSDN.

Also,

So, what's the reason to write that keyword, or why does it even exist?

Specifying this explicitly helps denote your intention to make the type private, very explicitly. This helps with maintainability of your code over time. This can help with other developers (or yourself) knowing whether a member is private by default or on purpose, etc.

AFAIK, private is the default everywhere in C#

Not quite - the default is "the most restricted access available for this declaration". So for example, with a top-level type the default is internal; for a nested type the default is private.

So, what's the reason to write that keyword, or why does it even exist?

It makes it explicit, which is good for two reasons:

• It makes it clearer for those who don't know the defaults, as per your question (I've never liked this argument, personally, but I figured it's worth mentioning)
• It gives an impression that you've deliberately decided to make it private, rather than just gone with the defaults.

As for your last part:

Moreover is there a case where writing "private" (alone) will change the accessibility of the member?

Yes, for making half of a property more restrictive than the other:

// Public getter, public setter
public int Foo { get; set; }

// Public getter, private setter
public int Bar { get; private set; }

I used to go with defaults everywhere I could, but I've been convinced (partly by Eric Lippert) that making it clear that you've thought about it and decided to make something private is a good idea.

Personally I wish there were a way of doing that for sealed / unsealed, too, for type declarations - possibly not even have a default. I suspect that many developers (myself included if I'm not careful) leave classes unsealed just because it's less effort than making them sealed.

private adds visual clutter. To those who insist that it makes things explicit, I would ask: Do you do this with mathematics, too? For instance:

var answer = a + b / c;

Do you find that unclear without redundant parentheses around b / c?

The rule in C# is very simple: By default, everything is as close to private as it can be. So if you need something to be more visible than the default, add a modifier. Otherwise, don't add needless keywords to your code.