SQL> create user alter_test identified by alter_test;

User created.

SQL> grant create session to alter_test;

Grant succeeded.

SQL> connect alter_test/alter_test
Connected.

SQL> alter user alter_test identified by alter_test;

User altered.

SQL> connect system/xxxxxx
Connected.
SQL> create or replace trigger alter_trigger
2 before alter
3 on database
4 declare
5 begin
6 if (ora_dict_obj_type = 'USER') then
7 raise_application_error(-20010,'you cannot change your own password');
8 end if;
9 end;
10 /

Trigger created.

SQL> connect alter_test/alter_test
Connected.
SQL> alter user alter_test identified by x;
alter user alter_test identified by x
*
ERROR at line 1:
ORA-00604: error occurred at recursive SQL level 1
ORA-20010: you cannot change your own password
ORA-06512: at line 4