Yes, a recursive CTE solves this rather nicely:

;WITH WGBizTree (bid, bname, wgid, wgname) AS

-- anchor part

(SELECT wb.bid, b.name AS bname, wb.wgid, w.name AS wgname

FROM users_workgroup uw

INNER JOIN workgroup_bizunit wb ON uw.wgid = wb.wgid

INNER JOIN bizunit b ON wb.bid = b.id

INNER JOIN workgroup w ON wb.wgid = w.id

WHERE uw.uid = @user-id

UNION ALL

-- recursive part

SELECT b.id, b.name AS bname, bt.wgid, w.name AS wgname

FROM WGBizTree bt

INNER JOIN bizunit b ON bt.bid = b.parent_id

INNER JOIN workgroup w ON bt.wgid = w.id)

SELECT bname, wgname

FROM WGBizTree

ORDER BY bname, wgname

The key is to have an anchor query that gets the business units directly associated to the workgroups that the specified user is in, then join itself back to the bizunit table by parent_id to get the children business units.