Need SQL outer join help with 3 tables
First, join TABLE_2 and TABLE_3 using an inner join and additionally filtering on Value_A = 'a':

SELECT
  t2.ID_1,
  t3.ID_A,
  t3.Value_A
FROM
  TABLE_2 t2
  INNER JOIN TABLE_3 t3 ON t2.ID_A = t3.ID_A
WHERE
  t3.Value_A = 'a'


This will give you the following result set:

ID_1  ID_A  Value_A
----  ----  -------
1     b     a
4     d     a


Now use the above as a derived table and join it, using an outer join this time, to TABLE_1, additionally filtering the results on Value_1 = 11:

SELECT
  t1 .ID_1,
  t23.ID_A,
  t1 .Value_1,
  t23.Value_A
FROM
  TABLE_1 t1
  LEFT JOIN
  (
    SELECT
      t2.ID_1,
      t3.ID_A,
      t3.Value_A
    FROM
      TABLE_2 t2
      INNER JOIN TABLE_3 t3 ON t2.ID_A = t3.ID_A
    WHERE
      t3.Value_A = 'a'
  ) t23 ON t1.ID_1 = t23.ID_1
WHERE
  t1.Value_1 = 11
;


That will give you the output you want:

ID_1  ID_A  Value_1  Value_A
----  ----  -------  -------
1     b     11       a
3     NULL  11       NULL


However, nesting a query is not the only way to solve your problem – you can also use a nested join, which is much more concise:

SELECT
  t1.ID_1,
  t3.ID_A,
  t1.Value_1,
  t3.Value_A
FROM
  TABLE_1 t1
  LEFT JOIN
    TABLE_2 t2
    INNER JOIN TABLE_3 t3 ON t2.ID_A = t3.ID_A AND t3.Value_A = 'a'
  ON t1.ID_1 = t2.ID_1
WHERE
  t1.Value_1 = 11
;


The last query implements exactly the same logic as the previous query: first, tables TABLE_2 and TABLE_3 are joined and the result is filtered, then it is joined to TABLE_1 and the final set is filtered again. Some people also add brackets around a nested join:

FROM
  TABLE_1 t1
  LEFT JOIN
  (
    TABLE_2 t2
    INNER JOIN TABLE_3 t3 ON t2.ID_A = t3.ID_A AND t3.Value_A = 'a'
  )
  ON t1.ID_1 = t2.ID_1


to make it clearer (perhaps both for themselves and for future maintainers) that the nested join takes place before the outer-level one, logically, although the syntax is unambiguous enough without them.

Nevertheless, many people find it confusing even with brackets, and if you find yourself struggling with it as well, there is another option – right outer join:

SELECT
  t1.ID_1,
  t3.ID_A,
  t1.Value_1,
  t3.Value_A
FROM
  TABLE_2 t2
  INNER JOIN TABLE_3 t3 ON t2.ID_A = t3.ID_A AND t3.Value_A = 'a'
  RIGHT JOIN TABLE_1 t1 ON t1.ID_1 = t2.ID_1
WHERE
  t1.Value_1 = 11
;


Again, the logical sequence of events specified by the FROM clause repeats that of the derived table solution: a join between TABLE_2 and TABLE_3, filtered, is followed by a join with TABLE_1. The different syntax does not alter the outcome and the results produced still match your requirements.