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db<>fiddle
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CREATE TABLE t1 (
ClientID int NOT NULL,
Service varchar(25) NOT NULL,
Status varchar(25) NOT NULL,
CloseDate date,
);


insert into t1 values
(1,'A','Closed','7-1-2020'),
(1,'A','Open',null),
(1,'A','Open',null),
(2,'A','Open',null),
(3,'A','Closed','7-2-2020')
5 rows affected
SELECT ClientID,
Service,
Status,
CloseDate,
ISNULL((SELECT TOP 1 COUNT(*) OVER(PARTITION BY clientid ORDER BY clientid) FROM t1 WHERE t1.ClientID = t.ClientID and t1.CloseDate IS NULL),0)
FROM t1 t;

ClientID Service Status CloseDate (No column name)
1 A Closed 2020-07-01 2
1 A Open null 2
1 A Open null 2
2 A Open null 1
3 A Closed 2020-07-02 0