All INNER JOINs and UNIONs maybe

PROPOSED QUERY

select id t1_id,NULL t2_id,NULL t3_id,NULL t4_id from t1
union select t1.id,t2.id,NULL,NULL from t1
    inner join t2 using (id)
union select t1.id,t2.id,t3.id,NULL from t1
    inner join t2 using (id)
    inner join t3 using (id)
union select t1.id,t2.id,t3.id,t4.id from t1
    inner join t2 using (id)
    inner join t3 using (id)
    inner join t4 using (id)
;

SAMPLE DATA

use test
drop table if exists t1;
drop table if exists t2;
drop table if exists t3;
drop table if exists t4;
create table t1
(
    id int not null,
    name varchar(20),
    primary key (id)
); 
insert into t1 values (1,'name1');
create table t2 like t1; insert into t2 values (1,'name2');
create table t3 like t1; insert into t3 values (1,'name3');
create table t4 like t1; insert into t4 values (1,'name4');

PROPOSED QUERY EXECUTED

mysql> select id t1_id,NULL t2_id,NULL t3_id,NULL t4_id from t1
    -> union select t1.id,t2.id,NULL,NULL from t1
    ->     inner join t2 using (id)
    -> union select t1.id,t2.id,t3.id,NULL from t1
    ->     inner join t2 using (id)
    ->     inner join t3 using (id)
    -> union select t1.id,t2.id,t3.id,t4.id from t1
    ->     inner join t2 using (id)
    ->     inner join t3 using (id)
    ->     inner join t4 using (id)
    -> ;
+-------+-------+-------+-------+
| t1_id | t2_id | t3_id | t4_id |
+-------+-------+-------+-------+
|     1 |  NULL |  NULL |  NULL |
|     1 |     1 |  NULL |  NULL |
|     1 |     1 |     1 |  NULL |
|     1 |     1 |     1 |     1 |
+-------+-------+-------+-------+
4 rows in set (0.00 sec)

mysql>

All the columns have id as a common name. If you want, I can adjust my answer with the id names

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.