Using array or set comparisons:
create table t (str text);
insert into t values ('AAA'), ('BBB'), ('DDD999YYY'), ('DDD099YYY');
select str from t
where str like any ('{"AAA%", "BBB%", "CCC%"}');
select str from t
where str like any (values('AAA%'), ('BBB%'), ('CCC%'));
It is also possible to do an AND which would not be easy with a regex if it were to match any order:
select str from t
where str like all ('{"%999%", "DDD%"}');
select str from t
where str like all (values('%999%'), ('DDD%'));