Accepting that in Mathematica -c/y is automatically converted to -1*c*y^-1 and permitting the result shown in Andy's answer I believe we can use a simpler approach, at least for the kind of expression given in example.

Define rules that determine how a Times or Power expression should be counted, then uses Cases to find all instances in you expression and total them with Tr:

rules = {
   _*x__ :> Length@{x},
   _^n_?Positive :> n - 1
 };

expr = a + s^2*b - c/y + o^3 + n*m*u;
    
Tr @ Cases[expr, #, -2] & /@ rules

Tr @ %

{5, 3}

8

As a single function:

fn[expr_] :=
 Tr @ Cases[expr, #, -2] & /@ {_*x__ :> Length@{x}, _^n_?Positive :> n - 1} // Tr

String conversion

If you prefer a string processing result, now accepting that the form Mathematica uses may seem rather arbitrary, I propose:

stringfn[expr_] := 
  StringCases[
    ToString[expr, InputForm],
    {"*" :> 1, "^" ~~ d__?DigitQ :> FromDigits[d] - 1}
  ] // Tr

a + s^2*b - c/y + o^3 + n*m*u // stringfn

6

Accepting that in Mathematica -c/y is automatically converted to -1*c*y^-1 and permitting the result shown in Andy's answer I believe we can use a simpler approach, at least for the kind of expression given in example.

Define rules that determine how a Times or Power expression should be counted, then use Cases to find all instances in you expression and total them with Tr:

rules = {
   _*x__ :> Length@{x},
   _^n_?Positive :> n - 1
 };

expr = a + s^2*b - c/y + o^3 + n*m*u;
    
Tr @ Cases[expr, #, -2] & /@ rules

Tr @ %

{5, 3}

8

As a single function:

fn[expr_] :=
 Tr @ Cases[expr, #, -2] & /@ {_*x__ :> Length@{x}, _^n_?Positive :> n - 1} // Tr

String conversion

If you prefer a string processing result, now accepting that the form Mathematica uses may seem rather arbitrary, I propose:

stringfn[expr_] := 
  StringCases[
    ToString[expr, InputForm],
    {"*" :> 1, "^" ~~ d__?DigitQ :> FromDigits[d] - 1}
  ] // Tr

a + s^2*b - c/y + o^3 + n*m*u // stringfn

6

Accepting that in Mathematica -c/y is automatically converted to -1*c*y^-1 and permitting the result shown in Andy's answer I believe we can use a simpler approach, at least for the kind of expression given in example.

Define rules that determine how a Times or Power expression should be counted, then uses Cases to find all instances in you expression and total them with Tr:

rules = {
   _*x__ :> Length@{x},
   _^n_?Positive :> n - 1
 };

expr = a + s^2*b - c/y + o^3 + n*m*u;
    
Tr @ Cases[expr, #, -2] & /@ rules

Tr @ %

{5, 3}

8

As a single function:

fn[expr_] :=
 Tr @ Cases[expr, #, -2] & /@ {_*x__ :> Length@{x}, _^n_?Positive :> n - 1} // Tr

Accepting that in Mathematica -c/y is automatically converted to -1*c*y^-1 and permitting the result shown in Andy's answer I believe we can use a simpler approach, at least for the kind of expression given in example.

Define rules that determine how a Times or Power expression should be counted, then uses Cases to find all instances in you expression and total them with Tr:

rules = {
   _*x__ :> Length@{x},
   _^n_?Positive :> n - 1
 };

expr = a + s^2*b - c/y + o^3 + n*m*u;
    
Tr @ Cases[expr, #, -2] & /@ rules

Tr @ %

{5, 3}

8

As a single function:

fn[expr_] :=
 Tr @ Cases[expr, #, -2] & /@ {_*x__ :> Length@{x}, _^n_?Positive :> n - 1} // Tr

String conversion

If you prefer a string processing result, now accepting that the form Mathematica uses may seem rather arbitrary, I propose:

stringfn[expr_] := 
  StringCases[
    ToString[expr, InputForm],
    {"*" :> 1, "^" ~~ d__?DigitQ :> FromDigits[d] - 1}
  ] // Tr

a + s^2*b - c/y + o^3 + n*m*u // stringfn

6