I let things get as ridiculous as possible, accepting a trade every time it was offered, and I ended up with 54 picks in total:
Chris Conley at 5:20, Ty Montgomery at 5:31, Stefon Diggs at 6:21 were all steals in my opinion.
Brock Hekking at 7:29 will have an NFL career. Will be interesting to see if he gets drafted though.
Round 5 Pick 12 (N.O.): B.J. Finney, C, Kansas State (A)
Round 5 Pick 17 (S.D.): Tyler Varga, FB, Yale (A-)
Round 5 Pick 20 (PHI): Chris Conley, WR, Georgia (A-)
Round 5 Pick 21 (CINN): Karlos Williams, RB, Florida State (C+)
Round 5 Pick 29 (IND): Ladarius Gunter, CB, Miami (Fla.) (C+)
Round 5 Pick 30 (G.B.): Robert Myers, OG/OT, Tennessee State (B)
Round 5 Pick 31: Ty Montgomery, WR, Stanford (C+)
Round 5 Pick 32 (T.B.): Ray Drew, DE, Georgia (A-)
Round 6 Pick 7 (CHI): Blake Bell, TE, Oklahoma (A-)
Round 6 Pick 8 (T.B.): Jeremiah Poutasi, OT/OG, Utah (C+)
Round 6 Pick 9 (ATL): Sean Mannion, QB, Oregon State (C+)
Round 6 Pick 10 (NYG): Zack Hodges, OLB, Harvard (A-)
Round 6 Pick 11 (N.O.): Kaleb Eulls, DT, Mississippi State (B)
Round 6 Pick 14 (S.F.): Adrian Amos, FS, Penn State (B+)
Round 6 Pick 16 (S.D.): Jordan Richards, SS, Stanford (C)
Round 6 Pick 18 (BUF): Justin Manton, K, Louisiana-Monroe (A)
Round 6 Pick 19 (HOU): Wes Saxton, TE, South Alabama (C)
Round 6 Pick 20 (PHI): Bryce Hager, ILB, Baylor (A)
Round 6 Pick 21 (CINN): Stefon Diggs, WR, Maryland (D-)
Round 6 Pick 22 (ARI): Shaquille Riddick, DE/OLB, West Virginia (D+)
Round 6 Pick 24 (DET): JaCorey Shepherd, CB, Kansas (B)
Round 6 Pick 25 (CAR): Hayes Pullard, ILB, Southern California (C-)
Round 6 Pick 26 (CLE): Martrell Spaight, OLB, Arkansas (C)
Round 6 Pick 27 (DEN): Trenton Brown, OT, Florida (F)
Round 6 Pick 28 (BALT): Leon Brown, OG, Alabama (F)
Round 6 Pick 29 (IND): Ellis McCarthy, DT, UCLA (D-)
Round 6 Pick 30 (G.B.): Zach Zenner, FB/RB, South Dakota State (D+)
Round 6 Pick 31 (IND): Kyle Loomis, P, Portland State (C+)
Round 6 Pick 32 (TENN): Malcolm Brown, RB, Texas (B-)
Round 7 Pick 1 (T.B.): Erick Dargan, SS, Oregon (D)
Round 7 Pick 2 (N.E.): Xzavier Dickson, OLB, Alabama (C-)
Round 7 Pick 3 (JAX): Josh Harper, WR, Fresno State (B-)
Round 7 Pick 4 (OAK): Austin Hill, WR, Arizona (C+)
Round 7 Pick 6 (NYJ): Greg Mancz, C, Toledo (D-)
Round 7 Pick 7 (NYJ): Nick Boyle, TE, Delaware (D+)
Round 7 Pick 8 (ATL): Quandre Diggs, CB, Texas (D+)
Round 7 Pick 10 (ATL): Tony Miliano, K, Cincinnati (F)
Round 7 Pick 11 (MINN): Zach Zwinak, FB, Penn State (F)
Round 7 Pick 12 (CLE): Junior Salt, OG, Utah (F)
Round 7 Pick 13 (N.O.): Nick Perry, FS, Alabama (F)
Round 7 Pick 14 (DET): Connor Halliday, QB, Washington State (D-)
Round 7 Pick 15 (MINN): Michael Dyer, RB, Louisville (F)
Round 7 Pick 18 (HOU): DaVaris Daniels, WR, Notre Dame (F)
Round 7 Pick 19 (DAL): Eric Tomlinson, TE, Texas-El Paso (F)
Round 7 Pick 20 (PHI): Justin Coleman, CB, Tennessee (F)
Round 7 Pick 21 (CINN): J.R. Tavai, OLB, Southern California (D-)
Round 7 Pick 22 (PITT): Casey Pierce, TE, Kent State (C-)
Round 7 Pick 24 (ARI): Dres Anderson, WR, Utah (D+)
Round 7 Pick 25 (CAR): Malcolm Agnew, RB, Southern Illinois (C)
Round 7 Pick 26 (DAL): Joey Mbu, DT, Houston (B-)
Round 7 Pick 27 (IND): Antwan Goodley, WR, Baylor (E)
Round 7 Pick 28 (NYG): Tevin McDonald, FS/SS, Eastern Washington (E)
Round 7 Pick 29 (S.F.): Brock Hekking, DE/OLB, Nevada (E)
Round 7 Pick 31: Vince Mayle, WR, Washington State (A)