An odd prime $p$ defines the sequence $\left(\frac{1}{p}\right),\left(\frac{2}{p}\right),\ldots,\left(\frac{p-1}{p}\right)$
of values of the Legendre symbol describing the quadratic nature 
of integers modulo $p$. We denote by $R(p)$ the longest run (largest subset of consecutive squares or of consecutive non-squares modulo $p$) of this sequence. Naively assuming the sequence to be similar
to a sequence of $p-1$ tosses of a fair coin, we should expect 
$R(p)$ to be of size $\log(p-1)/\log(2)$.
Since $\log(p)$ is the mean distance between $p$ and the following (or previous) prime, we should have
$$\frac{\log(2)}{n}\sum_{p\leq n}R(p)\longrightarrow 1$$
(where the sum is over all odd primes $\leq n$)
for $n\rightarrow \infty$.

This seems to be more or less the case numerically albeit
the quantity $\frac{\log(2)}{n}\sum_{p\leq n}R(p)$ is
strictly smaller than $1$ for the first few thousand initial values of $n$. This indicates a very slight bias for these sequences of Legendre symbols.

*Is this an artefact?*

A few complements: The set of primes with a given value of the longest run should be finite. Computations for primes up to $50000$
suggest the cardinalities 
$$1,2,2,8,6,27,30,70,125,254$$
for the sets of odd primes giving rise to longest runs of length $1,2,\ldots,10$.

First occurrences of 'records' have sometimes curious holes: The few smallest primes with maximal runs of length $29$ are smaller than
the smallest primes with maximal runs of length $25,26,27$ or $28$.