как-то так (синтаксис не проверял, но думаю идея будет понятна)
SELECT
f.app_id as app1
,s.app_id as app2
,count(*) as cnt
FROM app_technologies f
INNER JOIN app_technologies s ON s.technology_id = f.technology_id AND s.app_id <> f.app_id
GROUP BY f.app_id, s.app_id
ORDER BY cnt DESC
LIMIT 1