Drew Brees had the most games with 300+ passing yards, with 123 games.

who has the most career 300 passing yard games
Interpreted as:
Who has the most career 300 or more passing yard regular-season games?
BreesBradyManningRyanRiversRodgersRoethlisbergerMarinoFavreStaffordPalmerWarnerManningFoutsMoonCousinsRomoMahomesMontanaBledsoeElwayFlacco1231139376736968636257525251514946464139373635
NAMEGPYDSCMPATTPCTAVGYDS/GTDTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees12343,9673,5855,12969.98.6357.53286.41082.11741,201108.6
Tom BradyTom Brady11340,0963,2284,72868.38.5354.83026.4681.41911,177109.6
Peyton ManningPeyton Manning9332,4082,5633,80267.48.5348.52386.3982.6116790103.9
Matt RyanMatt Ryan7626,2962,2163,20169.28.2346.01544.8541.71591,083103.0
Philip RiversPhilip Rivers7325,5251,9692,90867.78.8349.71625.6762.6132770102.8
Aaron RodgersAaron Rodgers6923,8381,8322,71867.48.8345.51967.2281.01691,142114.5
Ben RoethlisbergerBen Roethlisberger6824,5631,9482,89467.38.5361.21595.5722.5153999101.5
Dan MarinoDan Marino6322,4301,6532,67361.88.4356.01676.2853.27660996.2
Brett FavreBrett Favre6221,0891,6482,53665.08.3340.11495.9632.599701100.1
Matthew StaffordMatthew Stafford5720,3571,5672,41165.08.4357.11405.8471.9110783102.7
Carson PalmerCarson Palmer5218,0371,4022,18464.28.3346.91115.1552.512286996.4
Kurt WarnerKurt Warner5217,9581,3812,03867.88.8345.31135.5623.0120799101.1
Eli ManningEli Manning5118,1351,4152,19264.68.3355.61145.2683.18052894.8
Dan FoutsDan Fouts5117,8461,2542,02062.18.8349.91085.3793.992.2
Warren MoonWarren Moon4917,2911,2832,07761.88.3352.91004.8703.410575890.3
Kirk CousinsKirk Cousins4616,0761,3061,84071.08.7349.51126.1301.693709111.1
Tony RomoTony Romo4615,5831,1851,77366.88.8338.81126.3482.786599104.2
Patrick MahomesPatrick Mahomes4114,9091,1531,68368.58.9363.61177.0311.860384111.6
Joe MontanaJoe Montana3913,8241,0601,56567.78.8354.51026.5442.873511105.3
Drew BledsoeDrew Bledsoe3712,8009951,59562.48.0345.9774.8483.08159791.1
John ElwayJohn Elway3612,1239091,45562.58.3336.8755.2362.58567595.7
Joe FlaccoJoe Flacco3511,7489751,54563.17.6335.7704.5291.96745393.6
Derek CarrDerek Carr3311,4638931,37165.18.4347.4695.0261.954299100.1
Andrew LuckAndrew Luck3311,6339271,49761.97.8352.5875.8392.66641894.6
Andy DaltonAndy Dalton3210,9668321,26565.88.7342.7705.5332.684548100.6
Jared GoffJared Goff
Kerry CollinsKerry Collins
Donovan McNabbDonovan McNabb
Trent GreenTrent Green
Vinny TestaverdeVinny Testaverde
Matt SchaubMatt Schaub
Boomer EsiasonBoomer Esiason
Jameis WinstonJameis Winston
Steve YoungSteve Young
Jay CutlerJay Cutler
Marc BulgerMarc Bulger
Mark BrunellMark Brunell
Jim EverettJim Everett
Jim KellyJim Kelly
Johnny UnitasJohnny Unitas
Ryan TannehillRyan Tannehill
Sonny JurgensenSonny Jurgensen
Carson WentzCarson Wentz
Ryan FitzpatrickRyan Fitzpatrick
Rich GannonRich Gannon
Matt HasselbeckMatt Hasselbeck
Justin HerbertJustin Herbert
Russell WilsonRussell Wilson
Cam NewtonCam Newton
Dak PrescottDak Prescott
StatMuse has game-level data for passing yards going back to the 1950 season.

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