Let $x_n$ be the smallest positive integer which is not a quadratic residue modulo any of the first $n$ odd primes. The question is: is there any bound on how quickly $x_n$ grows as a function of $n?$ Now, since the probability of being a residue modulo any given prime is very close to $1/2,$ by an independence heuristic, $x_n$ should grow roughly like $2^n.$ The problem is that independence clearly does not hold here: if you ask the "opposite" question: how big is $y_n,$ the smallest integer which is a square modulo all of the first $n$ odd primes, the answer is $y_n=1.$ When in doubt, experiment, and computing $\log x_n$ for the $n=2, \dotsc, 20$ gives the graph: In case you wonder, the slope is around $0.8$ (so a bit bigger than $\log 2,$ but $20$ is pretty small, so $x_n \sim 2^n$ seems like a reasonable guess. Does anyone have any nontrivial bounds, even mod "standard conjectures"?
EDIT Noam Elkies points out that a few more terms have been computed (I have gotten up to 24 in the meantime, but 27 is even better :)) The slope of the log is down to $0.76,$ so the conjecture that $\log(x_n) \sim n \log 2$ seems quite reasonable.
Further Edit In fact, if you ponder Noam's comment, you will note that the heuristic will give something like $\log n 2^n$ for the conjectured growth rate...
More Values Using some of joro's ideas, and a little more, we can push the list of values to 40.
{2, 2, 17, 17, 83, 167, 227, 398, 398, 5297, 64382, 69647, 116387,
214037, 214037, 430022, 5472953, 5472953, 8062073, 8062073, 8062073,
41941577, 86374763, 163520117, 163520117, 231912722, 231912722,
231912722, 545559467, 1728061733, 2832363203, 5638787822, 5638787822,
6154772762, 6154772762, 7012246247, 7012246247, 7012246247, 6571091781638, 7218195919667}