=INDEX(A1:C1,MATCH(MAX(A2:C2),A2:C2,0))

If there is more than one instance of MAX the formula will match the
leftmost instance.

Works great. It is likely that there WILL be more than one instance of
the max value in the range, but I can deal with that using a text
disclaimer or something.

I ended up adding another formula that uses an IF statement to display a
text message if the max value occurs in the range more than once, and
displays nothing otherwise. I formatted that cell in red so that it grabs
attention when the message appears.

--Tom