(Unintentionally I have previously asked a similar and perhaps in itself not uninteresting
question
http://mathoverflow.net/questions/48705/galois-theory-generalization-of-abels-theorem
but this is what I originally had in mind.)

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Let $L$ stand for the smallest extension of ${\Bbb Q}$ closed under the operation of adjoining all roots of all polynomials of the form $x^n+ax+b,a,b∈L$. 

What polynomials $p$ don't split over $L$?  In particular, how low
can one make the degree of such a $p$?  (This 
http://en.wikipedia.org/wiki/Bring%E2%80%93Jerrard_form#Bring.E2.80.93Jerrard_normal_form 
would seem to guarantee degree($p$) $> 5$.)

Classically, $S_n$ occurs as a Galois group for
certain $x^n+ax+b$, $n\geq 5$.  That means that
obstructions for $p$ splitting over such $L$
must reflect information beyond the Galois group
of $p$.  So absent a full answer to my question,
what candidates does one have for such an obstruction?
For example, does the form of the polynomial single
out particular representations of $S_n$?  

Again, absent a full answer, does the literature contain theorems about
polynomials not splitting over similar large extension of ${\Bbb Q}$?