CFG for words that are not a concatenation of the same word
The question has been asked before, but since it's faster to describe the answer than to find the previous question, I'll outline the answer here.

We can characterize words in $L$ as belonging to one of two classes:


Words of odd length.
Words $z_1 \ldots z_n w_1 \ldots w_n$ where $z_i \neq w_i$ for some $i$.


It is easy to generate words of the first type. As for words of the second type, for fixed $i$ and $n$ we can generate them using rules of the form $\Sigma^{i-1} a \Sigma^{n-i} \Sigma^{i-1} b \Sigma^{n-i} = \Sigma^{i-1} a \Sigma^{i-1} \Sigma^{n-i} b \Sigma^{n-i}$ and its counterpart with $a,b$ replaced by $b,a$. I'll let you figure out how to generate all of these words (for all $i,n$).