B737-800 Pilot reported recurring caution alerts; missing parts; and false documentation of aircraft maintenance resulted in unsafe aircraft operation.

Narrative

Pilots flying aircraft experienced a recurring PACK TRIP and MASTER CAUTION indications on rotation. This is an extremely critical time of flight; and necessitate a QRH procedure. This has happened 4 times since Day 0; Day 0; Day 1; Day 2; Day 3; and Day 7. The first two times it was written up; maintenance signed it off after a bite check was completed. The third time it happened; MEL (21-XX) was put in place. Then subsequently; over multiple days and bases; several parts were replaced and the MEL was removed on Day 6. The next day; on Day 7; PACK TRIP and MASTER CAUTION at rotation recurred. The maintenance action at this point was to initiate a DIFFERENT MEL (21-XA); which was then diagnosed as 'RT PACK VALVE FAN BYPASS VALVE MISSING'. In the sign off; there was no action documented to repair and replace the missing part; locate the missing part; or to test the system. Also; the new MEL reset the 'time-to-go' day count back to 10 days; which I consider grossly fallacious. I had never seen this particular MEL; nor had the person I spoke to Maintenance Control. Ultimately; the condition that causes the the Master Caution Light on rotation to illuminate likely still exists; as it has since Day 0. I believe it is unsafe and unprofessional to continue operation of this aircraft until it is repaired properly. Suggestions: Better training or oversight/leadership in maintenance assessment and troubleshooting. Prioritize was is right; not what is easy. More time and assets available to maintenance to facilitate above. Monitor/flag doubling up on MEL's. Whistleblower mechanism put in place to improve output; so that those who see flaws or errors in the system can communicate same without repercussion.