Is there a maximum Julian day? I have read various discussions of the Julian day, including that on the Wikipedia, but none of these appear to define the maximum Julian day or proleptic Gregorian year in which that maximum day would occur.

Is there a maximum Julian day? If so, what is it?
Since the Julian day is defined as a continuous count of seqential days starting at Jan 1 4013 BC, and there is no limit to the number of integers available for  the count, there is no specified maximum.

If you are asking when the Julian Day Number will no longer be useful, possible answers are when there is no one left to count it, or when it is replaced by another concept or definition. Note that there are already several shortened versions, such as Modified JD and Truncated JD, used when JDN became too large to fit in smaller computer word sizes with the desired precision.