SQL Group by MAX then get the MIN of all the groups
Depending on the DBMS, you can use LIMIT 1 or TOP 1 or FETCH FIRST 1 ROWS ONLY - in combination with ORDER BY MaxLogOut:

-- standard SQL, Postgres, SQL Server 2012+
SELECT 
    MAX(LogOutTime) AS MaxLogOut, id
FROM table1
GROUP BY id
ORDER BY MaxLogOut
    OFFSET 0 ROWS
    FETCH FIRST 1 ROWS ONLY ;

-- SQL Server
SELECT TOP (1)  
    MAX(LogOutTime) AS MaxLogOut, id
FROM table1
GROUP BY id
ORDER BY MaxLogOut ;

-- MySQL
SELECT 
    MAX(LogOutTime) AS MaxLogOut, id
FROM table1
GROUP BY id
ORDER BY MaxLogOut
    LIMIT 1 ;