You can use ANY with an array:
9.23.3. ANY/SOME (array)
expression operator ANY (array expression)
expression operator SOME (array expression)
The right-hand side is a parenthesized expression, which must yield an array value. The left-hand expression is evaluated and compared to each element of the array using the given operator, which must yield a Boolean result. The result of ANY is "true" if any true result is obtained. The result is "false" if no true result is found (including the case where the array has zero elements).
So you can say things like:
select first_name
from profiles
where first_name like any (array['on%', 'Ank%'])
You could also use a regex instead of LIKE:
first_name ~ '^(on|Ank)' -- Case sensitive
first_name ~* '^(on|Ank)' -- Case insensitive
|doesn't mean anything special with LIKE, you'd have to switch to SIMILAR TO or a regex and that might over complicate things (or not).