The MAX of two MAX values in SQL

The answer to your question is the GREATEST function. However, we must make a couple of small changes to your expressions to make the query work.

select greatest(
         ( select max(col1) from table1 )
       , ( select max(col2) from table2 )
               ) as col3 

Here each subquery produces a single value, and then GREATEST, which is a scalar function—as opposed to MAX, which is an aggregate function—selects the higher of those values.

If your particular database system does not support the standard SQL GREATEST function, use this:

select case when max1 > max2
            then max1
            else max2
        end as col3
  from ( select max(col1) as max1
           from table1 
       ) as t1
cross
  join ( select max(col2) as max2
           from table2 
       ) as t2 

Why a cross join? Because each of the derived tables (t1 and t2) returns only one row, so the cross join produces only one row, with two columns, max1 and max2.

Note that GREATEST can take any number of values. The CASE approach, however, becomes complex and cumbersome for more than two values.