I'm new at this, but I think the following formula does what you've asked.
=INDEX(A2:G2,MODE(MATCH(A2:G2,A2:G2,0)))
Your sheet is view only, so I can't place it there.
Replace my A2:G2 range in the formula with the range with your long list (either column or row) of values.
This will search the range for the amount of the most duplicates, and return the first value that is duplicated the most.
Note: it doesn't flag if there are other values that have an equal number of duplciates as the first value.
Here is a sample sheet, with examples using values in a row, or in a column:
https://docs.google.com/spreadsheets/d/1mIxTKXjED9kpqAGV55pf8Yjq102Sc5Ymn_yY6qh3XmE/edit?usp=sharing
Let me know if this doesn't achieve what you want.