What is the deeper meaning of renormalizability and how is it related to gauge invariance? I ask myself if the demand of local gauge invariance - say $U(1)$ invariance in free Dirac theory -$$L_D=\bar{\Psi}(i\gamma^\mu\partial_\mu-m)\Psi$$
is enough to define the full Maxwell-Dirac-Lagrangian UNIQUELY? I read that adding the Maxwell term $-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} $ demands for other requirements like 1) simplicity, 2) Lorentz invariance and 3) renormalizability. Similarly you need those other requirements for the interaction term $L_{int}=-j_\mu A^\mu$. While simplicity is clearly a metaphysical requirement Lorentz invariance is a genuinely physical one. But I am not sure about the role of renormalizability here?
What is the deeper physical meaning of it and why is the demand for renormalizability physically like the demand for Lorentz invariance?
 A: When you do naive calculations based on your Lagrangian (perturbation theory and thus Feynman diagrams, setting e.g. the parameter $m$ to the mass of the particle) you will at some point run into infinite results.
This is due to the fact, that $m$ in your Lagrangian (and other coupling constants) is not physical. The physical mass could for example be defined as the pole of the propagator, which will change with the order in perturbation theory.
Now these infinities can be erased by carefully defining what the physical parameters are and it then turns out, that the parameters of the Lagrangian (e.g. $m$)  cancel the infinities of your naive calculation.
At least this is so in a renormalizable theory. In a non-renormalizable theory it can happen, that at some point in your perturbation theory you would get infinities you could not get rid of, without introducing new terms in the Lagrangian.
However, their are certainly people who believe that renormalizability is not a fundamental requirement for a theory which we only use in perturbation theory and which we know to break down at some high energy. In that sense I would not say that you can compare the demand for renormalizability with the demand for Lorentz invariance in its physical inevitability. But it produces "nicer" theories, which could also be argued to be simpler.
