Why this weird return value from `N`?

As @t-smart seems to suspect, the change in V11.3 to let machine underflows to underflow to 0. is at the root of this bug. In the OP's problem, Mathematica isolates the roots with rational numbers to the desired precision. It then uses the interval length to determine the precision argument to N to get the final result. Unfortunately it uses machine precision to approximate the interval length, which underflows to 0.. The resulting precision, based on Log[0.] is Indeterminate, which is an invalid precision argument for N. This failure results in N returning unevaluated.

FWIW, a fix is to redefine N via the Villegas-Gayley trick:

Internal`InheritedBlock[{N},
 Unprotect[N];
 N[e_] /; ! TrueQ[$in] := Block[{$in = True},
   With[{ne = N[e]},
    If[e != 0 && ne == 0.,
     N[e, $MachinePrecision],
     ne
     ]
    ]];
 Protect[N];
 N[Root[{750 + 5*E^((Log[3] - Log[5] + Log[13])*#1) - 300*#1 - 
      2*E^((Log[3] - Log[5] + Log[13])*#1)*#1 - 
      5*E^((Log[3] - Log[5] + Log[13])*#1)*(Log[3] - Log[5] + 
         Log[13])*#1 + 
      E^((Log[3] - Log[5] + Log[13])*#1)*(Log[3] - Log[5] + 
         Log[13])*#1^2 &, 
    1.61954714423535277337584345129603991681`20.521825446421918}], 
  2010]
 ]
(*
1.61954714423535277337906541500279581776848110660831546798133301726783\
...\
317430074329935276876493295501167507324563338238323  
*)

I cannot explain the threshold precision (which seems to be Log10[2*^2003]) for the bug.

I suppose this is a problem involving the numeric precision with Mathematica. Since I don't have the source code, my best guess is that somehow the rules dealing with N changed in the past update. N[Khinchin, 2000] This code runs fine on Mathematica 11.2, giving a correct result. But on Mathematica 11.3, this is what it get: So in 11.3, MMA changed the rules and set a hard stop on the maximum digits. For your code, that is what happened, Root gets onto the hard stop, thus putting an end to the calculation. The reason of this change is unknown. Speed improvements, I suppose?