But you have, nevertheless, some misconceptions. First, understand what the different terms mean.
u is the applied voltage (it is a number fixed by the power supply/battery) not some "effective voltage"; iR is the voltage drop across the resistor; and Ldi/dt is the voltage drop across the ends of the inductor.
The fact that the electric field generated by the battery is a conservative field (ie : the work done by the field on a charge is path independent, or the work done over a closed loop is zero) requires that the voltage gain provided by the battery equal the total voltage drop across various elements in the circuit. This concept is contained within Kirchoff's Voltage Law.
At t=0, there is no current and hence no drop across the resistor. So, the entire potential drop is across the inductor, or Ldi/dt = u. Since u > 0, di/dt > 0. This tells you that i must increase from its initial value of 0, causing a potential drop to appear across the resistor. For the equation to work (which it must, for the reason stated in the previous paragraph), as iR increases, Ldi/dt must decrease, so as the keep the sum constant. Since Ldi/dt started out equal to u and has henceforth only decreased, it can not exceed u (in theory).
The equation never intends to give a physical reason why the current increases. The equation (as many others) merely tells you how to find the current at a certain time. The reason why the current increases is that the net force on the electrons is non-decreasing over timescales that are large compared to electron-phonon scattering times. In the absence of the inductor, when u=0, i=0; and as u instantaneously changes to some non-zero number (by throwing a switch), the current too reaches a non-zero value "instantaneously" and stays steady at that value. So, the current increases simply because of the appearnce of a driving force - the electric field generated by the battery. Thus, it is seen that the damping force from the resistor has an extremely short time (virtually zero) constant. This is indeed true and this time constant would be of the order of the relaxation time between scattering off the atoms in the resistor. This number is of the order of nanoseconds, and hence may be neglected as far as the big picture is concerned. This, however, is what you are talking about when you speak of the effective mass of the electron having a role in the damping (not the Ldi/dt term). Now you know why this contribution is not important to the behavior over longer time-scales.
When an inductor is included in the circuit, there is an addition retarding force coming from the induced magnetic field whose value is proportional to the rate of change of the field/current (and hence disappears in the steady state). This slows the current growth over time scales that are much bigger than the relaxation (or scattering) time.
From how you are using this term, it would be unphysical to have a "net voltage". This would require that the work done by the electric field over a closed loop be non-zero, resulting in a violation of energy conservation.