Here is the recursive CTE solution.

Not sure if it will be any quicker at all, and you still need to know the max iteration depth to put in an exit condition.

WITH

recursiveCTE AS (

SELECTorderid,

o.partykey,

p.personId,

1 AS incrementNbr

FROM #orders o

INNER JOIN #personIds p

ON p.partykey = o.partykey

UNION ALL

SELECT r.orderid,

p2.partykey,

p2.personId,

incrementNbr + 1 AS incrementNbr

FROM recursiveCTE r

INNER JOIN #personIds p1

ON p1.partykey = r.partykey

INNER JOIN #personIds p2

ON p2.personId = p1.personId

AND p2.partykey <> r.partykey

WHERE incrementNbr < 10

)

SELECT orderid,partykey,min(incrementNbr) AS nbrOfIterations

FROM recursiveCTE

GROUP by orderid,partykey

ORDER BY orderid, min(incrementNbr),partykey