Scientific notation for exact numbers?

You can try $Post or $PrePrint:

$Post

num = 37894580188800000000000000000000000000000000000000;

$Post = N; (* apply `N` to every output expression *)

num

3.78945801888`*^49

Precision[num]

∞

Reset

$Post = .

num

37894580188800000000000000000000000000000000000000

$PrePrint

Apply N to every expression before it is printed -- "Show exact quantities by their numerical value:"

$PrePrint = Replace[#, 
  x_?NumericQ :> If[Precision[x] == ∞, N[x], N[x, Precision[x]]], All] &;

num
foo[{num, bar[10`20], 2`2}]

You can create wrapper function to do this:

MakeBoxes[inexactForm[e_], StandardForm] ^:= Internal`InheritedBlock[{Integer},
    Unprotect[Integer];
    MakeBoxes[i_Integer, StandardForm] ^:= With[{n=N@i}, MakeBoxes[n, StandardForm]];
    MakeBoxes[e, StandardForm]
]

Then:

inexactForm[
    a[
        37894580188800000000000000000000000000000000000000,
        37894580188800000000000000000000000000000000000000
    ]
]

a[3.78946*10^49,3.78946*10^49]

You could just modify the display Format of specific functions/patterns:

Unprotect[BesselJ, BesselY];
Format[BesselJ[n_, z_ /; Precision[z] == ∞]] := BesselJ[n, N[z]]
Format[BesselY[n_, z_ /; Precision[z] == ∞]] := BesselY[n, N[z]]
Protect[BesselJ, BesselY];

then we get a more legible display without modifying any variables:

test

(*    (BesselJ[-(3/4), 5.40477*10^9 Mrest] BesselY[-(1/4), 5.40477*10^9 Mrest] + 
       BesselJ[3/4, 5.40477*10^9 Mrest] BesselY[1/4, 5.40477*10^9 Mrest]) (Quantity[...])    *)