Is there a string that's hash is equal to itself? I was wondering if there's any string that has a hash equal to itself, so that – when using any (none specific) hash function – the hash would be equal to that string?

so that:

hash(x) = x


Note that this is not an assignment or anything. I’m just curious and couldn't find any specific answer or reference. And I’m not sure how to go and prove/disprove that to myself!
I restrict to hash functions $H$ with an output of some fixed size $n\ge1$ bit(s), accepting as input some strings, including all $n$-bit strings; MD5 (resp. SHA-1, SHA-256) is an example of such function for $n=128$ (resp. $n=160$, $n=256$).
Whether there exists a solution to $H(x)=x$ depends on the particular hash function. If $H$ is a random function (as MD5, SHA-1, and SHA-256 aim to be), the answer is YES with odds next to $63.2\%$ for practical values of $n$.
More precisely: $H(x)=x$ can hold only if $x$ has exactly $n$ bits. There are $2^n$ values of $x$ that satisfy the later condition, and restricting $H$ to such $x$ there are $(2^n)^{(2^n)}$ different $H$ functions, of which $(2^n-1)^{(2^n)}$ such that $H(x)=x$ has no solution. Therefore, if we choose one $H$ uniformly randomly, odds are exactly $1-{(2^n-1)^{(2^n)}\over(2^n)^{(2^n)}}=1-(1-2^{-n})^{(2^n)}$ that we picked $H$ such that $H(x)=x$ has a solution. As $n$ increases, this converges very fast to $1-1/e\approx0.632$ (where $e\approx2.718$ is the base of the natural logarithm).
This does not tell if MD5 has the property that there exists a solution to $\operatorname{MD5}(x)=x$ (which would be a 128-bit bitstring $x$). The best we can say is that it likely holds, with odds about to 63%, but determining if the assertion is true or false is beyond our current computing power (the best method we have is exhaustive search, and if the answer is no it would require $2^{128}$ hashes; otherwise it is still likely to require over $2^{126}$ hashes, which is beyond reach).
PHP specific: if md5($string) === $string had some solution, that would be a 32-character string of hexadecimal lowercase characters; we are not hashing the same $2^{128}$ candidates as above so the question is not equivalent, but the reasoning can be adapted, and again the best we can say is that it is likely there's a solution, with odds about 63%.
Further, the original question asked if there is a string such that md5($string) == $string. To answer this, we must take into account how the == operator works in PHP due to type juggling (it holds that &quot;0042&quot; == &quot;42&quot;, and &quot;20e2&quot; == &quot;   +002000&quot;). It is overwhelmingly likely that there is a solution (just consider that among the $2^{200}$ strings consisting of 200 space or tab and an additional final 0, we expect about $31\cdot2^{72}$ hash to one of &quot;00000000000000000000000000000000&quot;, &quot;000000000000000000000000000000e0&quot; .. &quot;0e000000000000000000000000000000&quot;); however we can't exhibit one.

It is easy to define a hash function $H$ such that $H(x)=x$ has no solution: for example, define
$$H(x)=\begin{cases}x\oplus1&amp;\text{if }\operatorname{MD5}(x)=x\\\operatorname{MD5}(x)&amp;\text{otherwise}\end{cases}$$

It is also easy to define a hash function $H$ such that $H(x)=x$ has at least one solution: for example, choose some arbitrary 128-bit constant like $k=\text{af5d2bc6c9181f76f3161f43f41f6aeb}$, and define
$$H(x)=\begin{cases}k&amp;\text{if }x=k\\\operatorname{MD5}(x)&amp;\text{otherwise}\end{cases}$$

There can be no $x$ such that for all possible hash functions $H$, $H(x)=x$.
Proof by contraposition: assume there is such $x$, a function $H$ with $x$ having that property, and consider the function $\tilde H$ defined by $\tilde H(x)=H(x)\oplus1$.