If X+1=X , then what is the value of X?

Wait. Let's start with something easier, ok?

If x^4=16, what is x?

In some contexts, the answer is clear: x=2. In other contexts, the answer is just as clear: x=2 or x=-2. And in another context the obvious answer is x=2,-2,2i or -2i.

You can't definitively answer the question as written. Implicitly, the question is really this: “Consider the natural numbers (or the integers, or the reals, or the complex numbers, or the quaternions, or whatever.) Among those numbers, which are the ones whose fourth power is the number we call 16?”

Habit and experience made us sloppy. Textbooks and teachers give us equations to solve, and the domain of discourse is supposed to be clear from context. This is why most people would answer x^4=16 and z^4=16 differently, despite the fact that these two equations say the exact same thing.

Without a clear delineation of what precisely I'm looking for, simply saying “solve x^2=0" is meaningless. It's not a complete sentence. It's half a question. Am I asking about real numbers? Matrices? Elements of some ring? There are infinitely many answers to that question, all equally right and wrong.

The operation of squaring and the symbol 0 are overloaded. They mean different things in different domains, so asking you to solve this is like asking you “who is the captain?” What does that mean? What kind of captain? A maritime profession? A military rank? Where? When? Of what?

Ok. Now that we've established that, it may be clear that asking for “the value of x" given x+1=x is not a meaningful question. It's not up to me to figure out what sort of legitimate xs you have in mind, and what you mean by “+” or “1” or “=“. This is part of the question. If you don't tell me, you're asking me to make up a question, not answer one. I can make up many questions.

I may ask: if x is a natural number and x+1=x, what is x? And the answer is: there isn't any natural number satisfying this equation.

I may ask: if x is a real number and x+1=x, what is x? And the answer is: there isn't any real number satisfying this equation.

I may ask: if x is a cardinal number and x+1=x, what is x? And the answer is: x may be any infinite cardinal, for example \aleph_0. (Note that “+" now means something different than before.)

I may ask: if x is an ordinal number and x+1=x, what is x? And the answer is: there isn't any such ordinal number, but if you're ok with 1+x=x then any infinite ordinal will do.

I may ask: if x is a boolean variable, + means “OR”, 1 means “True”, and x+1=x, what is x? And the answer is: x is 1, or “True”. (Lots of commenters seem to be interested in this version, so I added it in).

I may ask: if “=" doesn’t mean “is equal to” but rather “assign what’s on the right to the variable on the left”, as it does in some programming languages like C or Python, and if you re-write the thing as “x=x+1", then what is x? And the answer is: it’s the result of applying the “+1” operator, whatever that is for the appropriate data type, to whatever value x had before that command was executed.

Finally, many people are explaining in the comments that x is “+inf” or “-inf” or \infty or +\infty or -\infty or something like that. The only mathematical contexts where this notation makes sense are the extended real line in real analysis (including +\infty and -\infty) and the Riemann sphere in complex analysis (with \infty alone). Once again, if this is the intended context, fine, but without specifying that this is the intended context, those answers are not more or less correct than any other one I’ve mentioned.

So, there's no “the” value of x under those conditions. The question is incomplete, and different ways of completing it yield different questions with different answers.