Robustness of non-context-free proof against trivial manipulation First, we state here a theorem that is well-known in computability theory:


  $L=\{xx\mid x\in\Sigma^*\}\notin CFL$ for every fixed $|\Sigma|\geq2$


And, the standard proof is using pumping lemma. At first sight, we rarely face the language $L$ exactly as defined above in a broader theoretical research. Instead, what might be more frequent is $$L'=\{\text{some-simple-manipulation}(x)x\mid x\in\Sigma^*\}$$ Perhaps, those simple manipulation are necessary for unambiguous encoding. For example, symbol doubling, prefix, postfix, etc. Those are quite frequent and unavoidable.

Now, the question is: Can the above theorem still hold under these simple manipulations?

Symbol doubling (as commented below) is a good starter, if one feels the question too broad.

Or even worse, there may be some simple manipulation that push $L$ down to CFL again.
Note $\left\{x^{\mathrm{R}}x\mid x\in\Sigma^*\right\}$ is context free, where $x^{\mathrm{R}}$ means the reverse of $x$. 


  Or even worse, there may be some simple manipulation that push $L$ down to CFL again.


So if you consider reversing as a simple manipulation, the answer is indeed yes.