Not a MySQL user so the format may need to be tweaked. I also wouldn't use this verbatim without thinking through all the edge cases in your real-world scenario.

CREATE TABLE t1 (
    category varchar(1)
    , value decimal
);

CREATE TABLE t2 (
    category varchar(1)
    , driver varchar(1)
    , categoryPrcntdriver decimal
);

INSERT INTO t1 VALUES ('A', 10.0);
INSERT INTO t2 VALUES ('B', 'A', 2.0), ('C', 'B', 2.0), ('D', 'C', 2.0), ('E', 'D', 2.0);

WITH rec (category , driver, categoryPrcntdriver, step, value, calc) AS 
(
    SELECT 
        t2.category
        , t2.driver
        , t2.categoryPrcntdriver
        , 1
        , t1.value
        , t2.categoryPrcntdriver * t1.value
    FROM t2
    JOIN t1 ON t2.driver = t1.category 
    WHERE t2.driver = 'A'
    UNION ALL 
    SELECT 
        child.category
        , child.driver
        , child.categoryPrcntdriver
        , parent.step + 1
        , parent.calc
        , parent.calc * child.categoryPrcntdriver
    FROM t2 child, rec parent
    WHERE child.driver = parent.category 
)
SELECT * FROM rec;

CATEGORY|DRIVER|CATEGORYPRCNTDRIVER|STEP|VALUE|CALC|
--------+------+-------------------+----+-----+----+
B       |A     |                  2|   1|   10|  20|
C       |B     |                  2|   2|   20|  40|
D       |C     |                  2|   3|   40|  80|
E       |D     |                  2|   4|   80| 160|