Moving your subselects to joins should yield some performance increase:
SELECT
c.Email,
MAX(c.Birthdate)
UPPER(c.Deleted) AS Deleted,
UPPER(c.Inactive) AS Inactive,
UPPER(c.SendEmail) AS SendEmail,
CONCAT('[',GROUP_CONCAT(DISTINCT c.Site SEPARATOR ']['),']') AS SITEID,
CONCAT('[',GROUP_CONCAT(DISTINCT c.Studio SEPARATOR ']['),']') AS STUDIO,
c2.SERVICES,
s1.`Date` as lastPurchaseFreeWeek,
NOW() as dateModified
FROM my_example c
INNER JOIN (
SELECT CONCAT('[',GROUP_CONCAT(DISTINCT s2.Service SEPARATOR ']['),']') AS SERVICES
FROM my_example c2
INNER JOIN my_example_sales s2 ON s2.idMember = c2.idMember AND s2.Site = c2.Site
) c2 ON c2.Email = c.Email
INNER JOIN (
SELECT MAX(date(s1.Date)) as `Date`
FROM my_example_sales s1
WHERE (CASE WHEN s1.Site = '1' THEN s1.ProductID IN ('1','6','7','12','18','22')
WHEN s1.Site = '2' THEN s1.ProductID = '156'
WHEN s1.Site = '3' THEN s1.ProductID IN ('3','5','6')
WHEN s1.Site = '4' THEN s1.ProductID IN ('11','15')
WHEN s1.Site = '5' THEN s1.ProductID = '23'
WHEN s1.Site = '6' THEN s1.ProductID = '23'
WHEN s1.Site = '7' THEN s1.ProductID = '23'
WHEN s1.Site = '8' THEN s1.ProductID = '23'
WHEN s1.Site = '9' THEN s1.ProductID = '23'
WHEN s1.Site = '10' THEN s1.ProductID IN ('7','11','17','30','31')
WHEN s1.Site = '11' THEN s1.ProductID = '23'
WHEN s1.Site = '12' THEN s1.ProductID IN ('7','11','17','30','31')
WHEN s1.Site = '13' THEN s1.ProductID = '23' END)
) s1 ON ( s1.idMember = c.idMember AND s1.Site = c.Site )
WHERE c.Email !=''
GROUP BY c.Email
ORDER BY c.ModDate DESC