B757 CREW WAS ASSIGNED 2 DIFFERENT STARS; THE 3 DIFFERENT RWYS ALL IN CLOSE PROX TO THE ARPT. THEN A FOREIGN ACR MISSED HIS TURN ONTO HIS ASSIGNED RWY AND BLUNDERED ACROSS THE PATH OF 3 OTHER RWYS. THE B757 HAD AN NMAC WITH THE FOREIGN ACR MD80.

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APCHING LAX ON CIVET ARR AT 1000 FT; APCH CTL CHANGED ARR TO STAR FOR RWY 24R. UPON CONTACT WITH SECOND APCH CTL; HAD APCH CHANGED TO ILS RWY 24L AND AT APPROX 8000 FT; CLRED FOR ILS RWY 24L. AT THIS STAGE WE WERE SOMEWHAT ABOVE THE GS BUT WITH ADEQUATE TIME TO MAKE AN EARLY DSCNT TO THE PROPER GLIDE PATH. AT APPROX 5000 FT; I SAW A FOREIGN ACR MD80 APCHING FROM THE L (S) AT A 90 DEG ANGLE. THE ACFT CONTINUED THROUGH OUR PATH AND PASSED ABOUT 500 FT BELOW US. THE ACFT CONTINUED N AND THEN E TO BE REVECTORED BY LAX APCH. LAX APCH ADVISED US THAT THE FOREIGN ACR HAD BEEN CLRED FOR A VISUAL TO RWY 25L; BUT HAD FLOWN THROUGH THE FINAL AND THUS INTO AND THROUGH OUR APCH PATH. THERE WAS A THIRD ACFT SLIGHTLY BEHIND US AND LOWER ON AN APCH FOR RWY 25R AND ANOTHER ON BASE FOR RWY 24R. FOR THIS REASON WE SELECTED TA; ON THE TCASII; AS TO AVOID NUISANCE RA'S DUE TO KNOWN TFC IN CLOSE PROX. WE WERE NOT TOLD OF THE FOREIGN ACR FLT OR OTHER NEAR ACFT UNTIL THE FOREIGN ACR HAD ALREADY CROSSED UNDER US. AFTER ADVISING US OF THE FOREIGN ACR INCURSION; THE APCH CTLR STATED 'WELCOME TO LOS ANGELES.' WHAT A WELCOME. OBSERVATIONS: 1) TOO MANY APCH CHANGES IN A VERY SHORT PERIOD OF TIME ACQUIRING MULTIPLE FMC/RAPID ADJUSTMENTS. WE HAD STAR/RWY 25L STAR/RWY 24L ILS RWY 24L. ALL THESE CHANGES CREATE TOO MANY OPPORTUNITIES FOR ERROR AND TOO MUCH IN COCKPIT WORK IN A HIGH DENSITY AREA. 2) APCH CTL OVERLOADED WITH TFC; THUS UNABLE TO GIVE INFO AS TO OTHER TFC IN AREA. 3) APCH USING MULTIPLE FREQS/SECTORS; THUS WE WERE TOTALLY UNAWARE OF OTHER AIRPLANES OPERATING VISUALLY TO LAX S COMPLEX OR ON ILS TO THE S COMPLEX. 4) HAD WE BEEN RIGHT ON THE GLIDE PATH INSTEAD OF A LITTLE HIGH THIS WOULD HAVE BEEN AN EXTREMELY CLOSE ENCOUNTER. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE FLT WAS COMING IN FROM THE E. THEY WERE INITIALLY CLRED TO RWY 25R ON A STAR ARR. IN A SPACE OF LESS THAN 5 MINS; THEY WERE RECLRED TO RWY 24R ON A DIFFERENT STAR; AND THEN THE ILS TO RWY 24L. IN AN AUTOMATED 'GLASS' AIRPLANE; ALL THE CHANGES MUST BE PROGRAMMED INTO THE FMC. THIS REMOVED 1 PLT FROM THE OP IN ORDER TO DO THE TYPING. 4 ACFT WERE TO DO PARALLEL APCHS: 1) FOREIGN ACR; MD80; VISUAL TO RWY 25L. 2) ACR DC9; VISUAL TO RWY 25R. 3) ACR ACFT; VISUAL TO RWY 24L. 4) RPTR'S ACFT; B757 ILS TO RWY 24R. BECAUSE OF THE HIGH TFC DENSITY; AND THE FACT THAT TCASII ALREADY SHOWED 2 KNOWN AMBER TARGETS; THE CREW SWITCHED TCASII TO TA ONLY. THIS IS PERMITTED IN THE ACR OPS MANUAL TO PREVENT NUISANCE TCASII WARNINGS. RPTR'S ACFT WAS HIGH ON THE GS. THIS WAS THE ONLY REASON THERE WAS NOT A MORE SERIOUS NMAC. INFLT VISIBILITY INTO THE SETTING SUN WAS POOR. THIS CAUSED A VERY HIGH COCKPIT WORKLOAD; TRYING TO KEEP THINGS SORTED OUT. THE RPTR THINKS TFC MUST NOT BE ALLOWED TO BECOME THIS HVY.