I think I had to kludge mine a bit (I’d still like to learn the MODEL clause). Anyway, here it goes:

SELECT
  id,
  min_date,
  max_date
FROM
  (
    SELECT
      id,
      start_date,
      end_date,
      connect_by_isleaf isleaf,
      level levelof,
      TO_DATE( SUBSTR( SYS_CONNECT_BY_PATH( TO_CHAR( start_date, 'MMDDYYYY' ), '-' ), 2, 8 ), 'MMDDYYYY' ) min_date,
      ( CASE WHEN CONNECT_BY_ISLEAF = 1 THEN end_date END ) max_date
    FROM
      (
        SELECT
          other_id,
          id,
          start_date,
          end_date,
          ( CASE
              WHEN start_date = LAG( end_date + 1 ) OVER ( PARTITION BY id ORDER BY end_date ) THEN
                LAG( other_id ) OVER ( PARTITION BY id ORDER BY end_date )
            END ) contiguous
        FROM
          (
            SELECT id, rownum other_id, start_date, end_date
            FROM transaction
          )
      )
      START WITH contiguous IS NULL
      CONNECT BY PRIOR other_id = contiguous
  )
WHERE isleaf = 1
  AND levelof > 1;

        ID MIN_DATE  MAX_DATE
---------- --------- ---------
         1 18-JAN-06 27-FEB-06
         1 08-APR-06 17-APR-06
         2 18-JAN-06 27-FEB-06
         2 08-APR-06 18-MAY-06