I just want to check that I understand the definition of a graded subring, and I haven't seen anyone write down what seems like the most obvious type of non-example for a graded subring.
Inside of $k[x]$ with the standard grading, surely you can take $k[x^2]\subseteq k[x]$ as a (non-graded) subring, where in $k[x^2]$ you could choose to let $x^2$ be in degree $1$, in which case this is not a graded subring.
As a random aside, I guess one might like to think of $k[x^n]$ as being $k[z]$ but with $z$ in degree $n$. (not what I do above for the non-example of course)
Well, it depends on what you mean by a "non-example." Another non-example of a graded subring is a banana. Presumably what you mean is an example of a subring which is not a graded subring.
In that case $k[x^2]$ does not work. With the grading inherited from $k[x]$, it is a graded subring. And changing the grading makes it not a subring at all (subrings inherit their grading from $k[x]$). Said another way, $k[x^2]$ with $x^2$ in degree $1$ admits a ring homomorphism to $k[x]$ but it is not a homomorphism of graded rings.
An example of a subring which is not a graded subring is $k[x^2 + x]$.