The theoretical maximum lift of a cylinder is 4π. At that point the forward and rear stagnation points coalesce on the bottom of the cylinder. Practical solutions use rotating cylinders (Flettner rotor) but will not reach this theoretical limit. There is a way in potential flow theory to view the flow around an airfoil as that around a cylinder (conformal mapping), which helps to see what the theoretical limits for airfoils are.
This means your maximum lift coefficient will never be higher than 12.5. Since BLC will mostly help with shifting the rear stagnation point (and a bit with the forward, too), it will achieve quite a bit less.
Blowing will allow you to shift the rear stagnation point most of the way to the bottom of that cylinder (draw a circular control surface around your wing and view it as that theoretical cylinder). When the forward stagnation point is ahead at the front and the rear at the bottom, lift (from potential theory) will be 2π (approx. 6.3).
Maximizing the lift coefficient always means to avoid flow separation during recompression from as much suction as possible, and to do this recompression in the smallest length possible. Blowing helps with that, as does adding new boundary layers by introducing gaps between airfoil elements.
Adding a slat and a slotted flap to a regular airfoil will bring you at most to a c$_{l_{max}}$ of 2.8. Using multi-element flaps helps, as demonstrated by the airfoil of the Boeing 727. It had a triple slotted flap which could reach a two-dimensional c$_{l_{max}}$ of 4.2 with 60° deflection and rearward movement of the flaps which increased the effective wing area. Note that wing sweep and ailerons reduced the practical maximum lift coefficient of the full aircraft to 3.0 (see plot below, taken from A. M. O. Smith's excellent article High-Lift Aerodynamics).
Another practical limit is the maximum Mach number in the suction peak of all the airfoil elements. Once this reaches 1.58, no lift growth could be observed in experiments. This translates to a maximum for the product of Mach squared and pressure coefficient of -1.0. In other words: You need to fly very slowly in order to achieve high values of c$_{l_{max}}$.
The only practical design I know which reached that magic limit of 6.3 given as the c$_{l_{max}}$ of its wing airfoil is the Antonov An-70. Here this number is valid for the inner wing which is immersed in the prop wash of the mighty Aerosila SV-27 propellers, driven by the 13,240 hp Progress D-27 turboprop engines. By using the flight speed and clean wing chord for the reference parameters, this looks to me a bit like cheating, but so is your blowing.
If you look from afar such that your wing becomes a point, the oncoming flow will bump straight into the wing (actually, it first rises up and hits the wing with the upwash angle), and leave the wing with a downwash angle which is twice as large as the upwash angle. In effect, your wing will impart a momentum to the flow which is the product of the mass flow through a circle with a diameter equal to the span of the wing, the initial flow speed and the sine of the upwash angle.
Adding this momentum to the air causes a reaction force on the wing which we split in lift and drag. Since this force vector is tilted aft more when the downwash angle gets larger, drag is nonzero and eats away a bit of the lift. But the reaction force will not become infinitely large, and even if it would, it still would have a vertical component, which then also would become infinitely large. Only when the reaction force points straight back will lift be limited.
In reality the downward speed component added by the wing is at most $$v_z = \frac{S\cdot c_L\cdot\frac{v^2}{2}\cdot\rho}{\frac{b^2}{4}\cdot\pi\cdot\rho\cdot v} = \frac{2\cdot c_L\cdot v}{\pi\cdot AR}$$
with $AR = \frac{b^2}{S}$ the aspect ratio of the wing. If we now insert that maximum lift coefficient from potential flow theory, we get $$v_z=\frac{8}{AR}\cdot v$$
With the aspect ratio of a typical airliner the downwash angle now would be 90°, making the angle by which the reaction force is tilted backwards 45°. This means lift is 70% of what the reaction force is - drag eats away some of the lift but still has a rather small effect. Still, L/D is only 1 at this point, so compared to normal flight drag is punishingly large.
