You can do this in three steps:

CREATE TABLE test2 AS
SELECT PersId 
FROM pers p
WHERE (
  chefID IS NOT NULL 
  OR gehalt < (
    SELECT MAX (
      gehalt * 1.05
    )
    FROM pers MA
    WHERE MA.chefID = p.chefID
  )
)

...

UPDATE pers P
SET P.gehalt = P.gehalt * 1.05
WHERE PersId
IN (
  SELECT PersId
  FROM test2
)
DROP TABLE test2;

or

UPDATE Pers P, (
  SELECT PersId
  FROM pers p
  WHERE (
   chefID IS NOT NULL 
   OR gehalt < (
     SELECT MAX (
       gehalt * 1.05
     )
     FROM pers MA
     WHERE MA.chefID = p.chefID
   )
 )
) t
SET P.gehalt = P.gehalt * 1.05
WHERE p.PersId = t.PersId

The problem is that, for whatever stupid reason, MySQL doesn't let you write a query like this:

UPDATE myTable
SET myTable.A =
(
    SELECT B
    FROM myTable
    INNER JOIN ...
)

That is, if you want to perform UPDATE/INSERT/DELETE operations on the table, you cannot query internally (but you Can reference fields in external tables...)

The solution is to replace the myTable instance in the subquery with (SELECT * FROM myTable) as shown below

UPDATE myTable
SET myTable.A =
(
    SELECT B
    FROM (SELECT * FROM myTable) AS something
    INNER JOIN ...
)

This will obviously cause the necessary fields to be implicitly copied into the temporary table, so this is allowed.