SQL Server 2019

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CREATE TABLE TEST (StartDate date, EndDate date);
INSERT INTO TEST
SELECT '8/20/2017', '8/21/2017'  UNION ALL
SELECT '8/22/2017', '9/22/2017'  UNION ALL
SELECT '8/23/2017', '9/23/2017'  UNION ALL 
SELECT '8/24/2017', '8/26/2017'  UNION ALL 
SELECT '8/28/2017', '9/19/2017'  UNION ALL 
SELECT '9/23/2017', '9/27/2017'  UNION ALL 
SELECT '9/25/2017', '10/10/2017' UNION ALL
SELECT '10/17/2017','10/18/2017' UNION ALL 
SELECT '10/25/2017','11/3/2017'  UNION ALL 
SELECT '11/3/2017', '11/15/2017';
10 rows affected

select *
from test


StartDate
EndDate


2017-08-20
2017-08-21

2017-08-22
2017-09-22

2017-08-23
2017-09-23

2017-08-24
2017-08-26

2017-08-28
2017-09-19

2017-09-23
2017-09-27

2017-09-25
2017-10-10

2017-10-17
2017-10-18

2017-10-25
2017-11-03

2017-11-03
2017-11-15


select min(startdate), max(enddate)
from (select t.*,
             sum(case when prev_enddate >= startdate then 0 else 1 end) over (order by startdate) as grp
      from (select t.*,
                   max(enddate) over (order by startdate rows between unbounded preceding and 1 preceding) as prev_enddate
            from test t
           ) t
     ) t
group by grp
order by min(startdate)


(No column name)
(No column name)


2017-08-20
2017-08-21

2017-08-22
2017-10-10

2017-10-17
2017-10-18

2017-10-25
2017-11-15

StartDate	EndDate
2017-08-20	2017-08-21
2017-08-22	2017-09-22
2017-08-23	2017-09-23
2017-08-24	2017-08-26
2017-08-28	2017-09-19
2017-09-23	2017-09-27
2017-09-25	2017-10-10
2017-10-17	2017-10-18
2017-10-25	2017-11-03
2017-11-03	2017-11-15