Mathematica unable to solve easy differential equation $y''+y\,y'=0$ with initial conditions $y(0)=1, y'(0)=-1$

It's well known, that DSolve is relatively weak - at least as of now. Frequently we need to help it a bit to obtain the desired solution. In your case, we just need to find the general solution first and then solve for the constant:

generalsol = DSolve[{y''[x] + y[x] y'[x] == 0}, y, x][[1]]

const = Solve[{y[0] == 1, y'[0] == -1} /. generalsol // TrigToExp, {C[1], C[2]}, 
   Method -> Reduce][[1]]

y[x] /. generalsol /. const // Simplify
Simplify[%, C[3] ∈ Integers]
(* Cot[1/4 (π + 2 x)] *)

DSolve as of V11, sets the Method option of Solve to Restricted. This probably fixes some things, but it gets in the way here. We can try the Villegas-Gayley trick to override the setting. This does it for all instances of Solve used by DSolve, but it's hard to target the instance that is needed.

Internal`InheritedBlock[{Solve},
 Unprotect[Solve];
 Solve[eq_, v_, opts___] /; ! TrueQ[$in] := Block[{$in = True},
   Solve[eq, v, Method -> Automatic, opts]
   ];
 Protect[Solve];
 sol = DSolve[{y''[x] + y[x] y'[x] == 0, y[0] == 1, y'[0] == -1}, y[x], x]
 ]
(*
  {{y[x] -> 
     ConditionalExpression[I Tanh[1/2 (I x + 1/2 I (-π + 8 π C[3]))], C[3] ∈ Integers]}}
*)

This can be simplified as @xzczd does, Simplify[sol, C[3] ∈ Integers].

Comment

Maples dsolve is able to solve this IVP without any fuss,

restart:
ode:=diff(y(x),x$2)+y(x)*diff(y(x),x$1)=0;
ics:=y(0)=1,D(y)(0)=-1;
sol:=dsolve({ode,ics},y(x));

plot(rhs(sol),x=0..3);