Left Joining Tables with tables that have subquery aggregate
SELECT per.personID,
       id.lastName,
       id.firstName,
       id.middleName,
       pos.StartDate,
       cal.[name],
       en.grade,
       pos.eligibility
FROM dbo.Person per
     INNER JOIN dbo.[Identity] id ON per.personID = id.personID
     INNER JOIN dbo.Enrollment en ON per.personID = en.personID
     INNER JOIN dbo.Calendar cal ON en.calendarID = cal.calendarID
     OUTER APPLY
(
    SELECT TOP 1 p.eligibility,
                 StartDate = CONVERT( VARCHAR(10), p.startDate, 101)
    FROM dbo.POSEligibility AS p
    WHERE p.personID = per.personID
    ORDER BY p.startDate DESC
) AS pos
WHERE en.grade = '01'
      AND cal.[name] LIKE '%BES'
      AND en.active = '1'
      AND en.endYear = '2017';


Note that the filter on cal.[name] is not sargable. This means that even if that column is indexed, the index will not be used for a seek due to the leading "%" in the predicate.