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?
BreesBradyManningRyanRiversRodgersRoethlisbergerMarinoFavreStaffordPalmerWarnerManningFoutsMoonRomoCousinsMontanaBledsoeElwayFlaccoLuck1231089374736968636256525251514946433937363533
NAMEGPYDSCMPATTPCTAVGYDS/GTDTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees12343,9673,5855,12969.98.6357.53286.41082.11741,201108.6
Tom BradyTom Brady10838,2913,0644,49168.28.5354.52916.5661.51831,116109.9
Peyton ManningPeyton Manning9332,4082,5633,80267.48.5348.52386.3982.6116790103.9
Matt RyanMatt Ryan7425,5512,1473,10669.18.2345.31494.8531.71561,054102.8
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 Stafford5620,0491,5392,36965.08.5358.01395.9461.9105748103.0
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
Tony RomoTony Romo4615,5831,1851,77366.88.8338.81126.3482.786599104.2
Kirk CousinsKirk Cousins4314,8341,2111,69571.48.8345.01056.2261.579626112.3
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
Andrew LuckAndrew Luck3311,6339271,49761.97.8352.5875.8392.66641894.6
Patrick MahomesPatrick Mahomes3211,7068941,30568.59.0365.8957.3211.647295114.1
Derek CarrDerek Carr3211,1568701,33465.28.4348.6675.0261.95329099.9
Andy DaltonAndy Dalton3110,6058021,21865.88.7342.1665.4302.584548101.0
StatMuse has game-level data for passing yards going back to the 1950 season.