How to get all roles that a user is a member of (including inherited roles)?

You can query the system catalog with a recursive query, in particular pg_auth_members:

WITH RECURSIVE cte AS (
   SELECT oid FROM pg_roles WHERE rolname = 'maxwell'

   UNION ALL
   SELECT m.roleid
   FROM   cte
   JOIN   pg_auth_members m ON m.member = cte.oid
   )
SELECT oid, oid::regrole::text AS rolename FROM cte;  -- oid & name

The manual about the cast to object identifier type regrole.

BTW 1: INHERIT is the default behavior of CREATE ROLE and doesn't have to be spelled out.

BTW 2: circular dependencies are not possible. Postgres disallows that. So we don't have to check for that.

This is a simplified version of Craig Ringer's answer that a non superuser can use directly:

 SELECT oid, rolname FROM pg_roles WHERE
   pg_has_role( 'maxwell', oid, 'member');

pg_roles is essentially a view on pg_authid accessible to public, as it doesn't reveal passwords, contrary to pg_authid. The base oid is even exported into the view. When not needing passwords, there's no point in creating the dedicated superuser-owned function.

Short version:

SELECT a.oid 
FROM pg_authid a 
WHERE pg_has_role('maxwell', a.oid, 'member');

Here we use a version of pg_has_role that takes a role name as the subject and role oid to test for membership, passing member mode so we test for inherited memberships.

The advantage of using pg_has_role is that it uses PostgreSQL's internal caches of role information to satisfy membership queries quickly.

You might want to wrap this in a SECURITY DEFINER function, since pg_authid has restricted access. Something like:

CREATE OR REPLACE FUNCTION user_role_memberships(text)
RETURNS SETOF oid
LANGUAGE sql
SECURITY DEFINER
SET search_path = pg_catalog, pg_temp
AS $$
SELECT a.oid 
FROM pg_authid a 
WHERE pg_has_role($1, a.oid, 'member');
$$;

REVOKE EXECUTE ON FUNCTION user_role_memberships(text) FROM public;

GRANT EXECUTE ON FUNCTION user_role_memberships(text) TO ...whoever...;

You can use pg_get_userbyid(oid) to get the role name from the oid without the need to query pg_authid:

SELECT a.oid AS member_oid, pg_get_userbyid(oid) AS member_name
FROM pg_authid a 
WHERE pg_has_role('maxwell', a.oid, 'member');