It is proved that a recurrent neural net with rational weights can be a super-Turing machine. Can we achieve this in practice ?
I presume the proof the OP is referring to can be found in this monograph by Hava Siegelmann?
In his article 'The Myth of Hypercomputation', the eminent computer scientist Martin Davis explains (p8-9) that there is nothing 'super Turing' about this formulation.
EDIT: It's looking like the claim about rational weights being super-Turing is made in this more recent paper by Siegelmann, which introduces an additional assumption of plasticity, i.e. that weights can be dynamically updated.
You mean real numbered weights (specifically, irrational). This would require a machine that has unlimited precision over irrational values. I've seen machine parts that have many qualities. I've never seen one that has unlimited qualities. QM may give us some magical transistors that can hold an unlimited number of different values - or by deferring computation into the future and then teleporting the results back into the past (our present). Outside of that, for classical systems, you'd need a analog device that can output irrational values with unlimited precision. I don't think we've discovered any devices that can do that.