The correct word to describe something that cannot be corrected is incorrigible.
\begin{itemize}
\item incorrectible: This refers to something that cannot be made correct, but it's not the most common word used in this context.
\item incorrectable: This is a form that is rarely used; the correct word should focus on the inability to correct something permanently.
\item uncorrected: This simply means that something has not been corrected yet, which is not the intended meaning.
\item incorrigible: This is the correct choice, as it directly refers to something that cannot be corrected or improved, typically used in contexts involving behaviors or conditions that are beyond fixing.
\end{itemize}
\boxed{\text{incorrigible}}