Tom Brady has played the most seasons with 20+ carries, with 19 seasons.

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most seasons with 20 rush attempts
Interpreted as:
most seasons by a player with 20 or more rush attempts
BradyGoreElwayAllenTarkentonPetersonBreesSmithRodgersRoethlisbergerFavreBynerRigginsPerryRyanVickManningTaylorBettisThomasYoungPaytonHarris1916161616151515141414141414131313131313131313
NAMEGPSEASATTRUSH YDSYPCRUSH YDS/GRUSH TDREC YDSRECTRGAVGREC YDS/GREC TD
Tom BradyTom Brady299196361,0431.63.525653321.70.20
Frank GoreFrank Gore241163,73516,0004.366.4813,9854846968.216.518
John ElwayJohn Elway234167743,4074.414.63361320.30.31
Marcus AllenMarcus Allen222163,02212,2434.155.11235,4115879.224.421
Fran TarkentonFran Tarkenton223166443,5605.516.030-120-0.10
Adrian PetersonAdrian Peterson183153,21914,9024.681.41192,4743054238.113.56
Drew BreesDrew Brees231154367381.73.22167789.60.31
Emmitt SmithEmmitt Smith226154,40918,3554.281.21643,2245151306.314.311
Aaron RodgersAaron Rodgers201146653,3005.016.434-1133-3.7-0.10
Ben RoethlisbergerBen Roethlisberger205144631,3182.86.418-922-4.50.00
Brett FavreBrett Favre223145241,7163.37.714-510-5.00.00
Earnest BynerEarnest Byner211142,0958,2613.939.2564,6055129.021.815
John RigginsJohn Riggins175142,91611,3523.964.91042,0902508.411.912
Joe PerryJoe Perry156141,7378,3784.853.7531,7962417.511.58
Matt RyanMatt Ryan200134511,3753.06.9125115.00.01
Michael VickMichael Vick143138736,1097.042.7361220.50.00
Peyton ManningPeyton Manning208133736061.62.918-211-2.00.00
Fred TaylorFred Taylor153132,53411,6954.676.4662,3842903488.215.68
Jerome BettisJerome Bettis192133,47913,6623.971.2911,4492001007.27.53
Thurman ThomasThurman Thomas182132,87712,0744.266.3654,458472299.424.523
Steve YoungSteve Young160136964,0235.825.143221.00.00
Walter PaytonWalter Payton190133,83816,7264.488.01104,5384929.223.915
Franco HarrisFranco Harris173132,94912,1204.170.1912,2873077.413.29
Bill BrownBill Brown180131,6305,7973.632.2523,14228111.217.523
Matthew StaffordMatthew Stafford173123651,2243.47.113-323-1.50.00
StatMuse has season-level data for rush attempts going back to the 1932 season.