It works for me too.  Good work!

I think this query solution works too:

Sub CreateServices()
  With CurrentProject.Connection
     .Execute _
     "CREATE TABLE Services" & _
     " (client_id VARCHAR(10) NOT NULL" & _
     ", service_date DATETIME NOT NULL" & _
     ", admission_date DATETIME NULL" & _
     ", PRIMARY KEY (client_id,service_date));"
     
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00044', #12/22/2006#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00044',  #12/24/2006#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00044',  #01/09/2007#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00044',  #01/10/2007#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00044',  #04/18/2007#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00122',  #10/03/2006#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00122',  #10/04/2006#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00989',  #03/07/2007#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00989',  #03/08/2007#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00989',  #03/09/2007#);"
     .Execute "INSERT INTO Services (client_id, service_date)" & _
     " VALUES ('00989',  #04/01/2007#);"
  End With
End Sub

Query: Preceded by Gap
------------------------
SELECT Services.client_id,
      Services.service_date
FROM   Services
WHERE  (((EXISTS (SELECT s.service_date
                 FROM   Services AS s
                 WHERE  Services.client_id = s.client_id
                        AND s.service_date = DATEADD('d',-
1,services.service_date))) = False));

Query: Get Admission Dates
---------------------------
SELECT   Services.client_id,
        Services.service_date,
        MAX([Preceded by Gap].[service_date])  AS Admission
FROM     Services
        INNER JOIN [Preceded by Gap]
          ON ([Preceded by Gap].[service_date] <= services.service_date)
             AND (Services.client_id = [Preceded by Gap].client_id)
GROUP BY services.client_id,services.service_date;

Query: Update Services Admissions
----------------------------------
UPDATE Services      
SET    Services.admission_date = dlookup("Admission","Get Admission Dates",
"client_id = '" & Services.client_id & "' AND service_date = #" &
Services.service_date & "#");

not a query, but ..

Sub UpdateAdmissions()
 
  Dim d As Date
  Dim s As String
  Dim dbs As DAO.Database
  Dim rst As DAO.Recordset
     
  Set dbs = CurrentDb()
 
  ' Note: for some reason I cannot do this in a query due to
  ' Operation must use an updateable query error.
 
  s = "SELECT client_id, Min(service_date) AS AdmitDate" & _
     " FROM (Select s.service_date, s.client_id FROM Services AS s" & _
     " INNER JOIN Services as s1 ON s.client_id = s1.client_id" & _
     " AND DateDiff('d',s.service_date, s1.service_date) < 2" & _
     " GROUP BY s.client_id, s.service_date) AS a" & _
     " GROUP BY client_id"
 
  ' open recordset using Number Workdays query
  Set rst = dbs.OpenRecordset(s, dbOpenForwardOnly)
  With rst
     ' loop through recordset and updating admission_date column
     ' in Services table with AdmitDate from the query
     Do
        s = "UPDATE Services SET Services.admission_date = #" & .Fields(1) & _
        "# WHERE Services.client_id= '" & .Fields(0) & "'"
        'Debug.Print s
        dbs.Execute s, dbFailOnError
        .MoveNext
     Loop Until .EOF
  End With
  rst.Close
  Set rst = Nothing
  Set dbs = Nothing
End Sub

So it seems that ClientID is a text field.  That is why you needed the
single quotes.

Is discharge date the last date in the series where there is no follow on
date?  If so, I would again try a two-query approach.  The first query would
set the discharge date on the last record.

UPDATE TheTable
SET TheTable.[DISCHARGE Date] =[Service Date]
WHERE Not Exists
   (SELECT *
    FROM TheTable as TMP
    WHERE Tmp.ClientID = TheTable.ClientID
     AND Tmp.ServiceDate = TheTable.Service Date +1)

Now that we have accomplished that, you could try something like the
following.
UPDATE TheTable
SET TheTable.[DISCHARGE Date] =
DMin("[DISCHARGE Date]","TheTable","ClientID='" & Clientid & "' AND [Service
Date] >=#" & [Service Date] & "#")
WHERE TheTable.[DISCHARGE Date] is Null

AGAIN, back up your data before you try this.