How to determine if a regular language L* exists
$L^*$ always exists. ${}^*$ is an operation on languages, just like ${}^2$ is an operation on numbers&nbsp;&ndash; whenever you have a number&nbsp;$x$, there is a number&nbsp;$x^2$ and, similarly, whenever you have a language&nbsp;$L$, there is a language&nbsp;$L^*$.  Specifically, $L^*$ is the language of all strings that can be made by concatenating zero or more strings from&nbsp;$L$.  That's well-defined for any language&nbsp;$L$.

What you've observed is that $L^*$ might have different properties from&nbsp;$L$.  If we take your example of $L = \{y,yx,xy\}$, we see that no string in&nbsp;$L$ has $xx$ as a substring ($L$&nbsp;is not the only language with this property but that's beside the point).  On the other hand, $L^*$&nbsp;includes the string&nbsp;$yxxy$, which does have&nbsp;$xx$ as a substring. That doesn't mean that $L^*$&nbsp;doesn't exist; it just means that it doesn't have some property that&nbsp;$L$ has. Well, that's no surprise because $L$ and&nbsp;$L^*$ are often different languages, in which case they can't have exactly the same properties.