What makes the self-inductance in a series LR circuit to fail stopping current forever?
The self-inductance is fighting the current's will to go through the circuit, and there comes a time that the self-inductance loses the fight if we wait long enough ...


First, it's just a silly analogy to say the current has a "will" or that the inductor is "fighting" it. This might help you visualize or remember what's the behavior going to be, but it has nothing to do with the physics of the circuit.


  ... and eventually the current reaches to the maximum value.


A more rigorous way to say this is, "in the limit as time goes to infinity, the current approaches a limiting value" 

$$\lim_{t\to\infty}i(t) = I_{max}$$

or "for any positive finite error \$\epsilon\$, if you wait long enough, the difference between the current and the limiting current \$I_{max}\$ will be less than \$\epsilon\$."


  I mean why doesn't its induced opposing voltage stay constant forever and stop any current flow? 


Because it's not the value of the current that induces the back-emf of the inductor, it's the rate of change of current.

If the current were fixed at 0 (for example), then the inductor would produce no back-emf. Thus the full source emf would be applied across the resistor. Which would produce a current through the resistor. Which must also flow through the inductor, contradicting our assumption that the current through the inductor is zero.