Drew Brees had the most games with 3+ passing touchdowns, with 97 games.

who has the most games with 3 touchdown passes
Interpreted as:
Who has the most regular-season games with 3 or more touchdown passes?
BreesBradyManningRodgersFavreMarinoRiversRoethlisbergerManningRyanStaffordTarkentonRomoMontanaUnitasWilsonDawsonFoutsBlandaYoungPalmerWarnerEsiason9793937572626154494542414040393838373736353434
NAMEGPTDCMPATTPCTYDSAVGYDS/GTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees973422,6203,70770.732,0558.6330.59.2551.5126852121.6
Tom BradyTom Brady933212,3343,40668.529,5388.7317.69.4441.3137842121.4
Peyton ManningPeyton Manning933272,2853,28369.628,5858.7307.410.0641.982530121.4
Aaron RodgersAaron Rodgers752621,7922,60668.822,5408.6300.510.1230.91531,013125.3
Brett FavreBrett Favre722441,6492,50465.920,1348.0279.69.7492.0100610114.8
Dan MarinoDan Marino622141,4532,34562.019,5698.3315.69.1753.267492105.6
Philip RiversPhilip Rivers611921,4422,10568.518,1588.6297.79.1472.289507116.2
Ben RoethlisbergerBen Roethlisberger541861,3531,98668.116,9158.5313.29.4301.5101674119.3
Eli ManningEli Manning491641,1601,75166.214,3668.2293.29.4472.770484111.5
Matt RyanMatt Ryan451481,1281,61969.713,9128.6309.29.1261.678511119.7
Matthew StaffordMatthew Stafford421421,0641,64864.613,3448.1317.78.6342.190566109.7
Fran TarkentonFran Tarkenton411366901,15459.810,2838.9250.811.8403.5113.9
Tony RomoTony Romo401328971,30668.711,9449.1298.610.1251.959378123.1
Joe MontanaJoe Montana401329071,34167.612,4169.3310.49.8312.3120.2
Johnny UnitasJohnny Unitas391346761,10661.110,7479.7275.612.1423.8117.3
Russell WilsonRussell Wilson381348091,17568.910,4358.9274.611.4161.485588128.8
Len DawsonLen Dawson3812751786260.08,76210.2230.614.7354.1117.1
Dan FoutsDan Fouts371278671,38162.811,8338.6319.89.2443.2107.5
George BlandaGeorge Blanda371326121,20850.710,0548.3271.710.9665.592.6
Steve YoungSteve Young361177521,11467.510,4559.4290.410.5252.261297123.1
Carson PalmerCarson Palmer351178181,20268.110,1798.5290.89.7373.161423113.7
Kurt WarnerKurt Warner341128011,15869.210,6959.2314.69.7332.859362118.6
Boomer EsiasonBoomer Esiason341117201,14862.710,1408.8298.29.7312.762502112.1
Sonny JurgensenSonny Jurgensen341176571,09260.210,1769.3299.310.7454.1109.6
John ElwayJohn Elway331106841,08862.99,0388.3273.910.1272.544349112.4
StatMuse has game-level data for passing touchdowns going back to the 1950 season.

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