A Boolean function that is not constant on affine subspaces of large enough dimension I'm interested in an explicit Boolean function $f \colon \\{0,1\\}^n \rightarrow \\{0,1\\}$ with the following property: if $f$ is constant on some affine subspace of $\\{0,1\\}^n$, then the dimension of this subspace is $o(n)$. 

It is not difficult to show that a symmetric function does not satisfy this property
by considering a subspace $A=\\{x \in \\{0,1\\}^n \mid x_1 \oplus x_2=1, x_3 \oplus x_4=1, \dots, x_{n-1} \oplus x_n=1\\}$. Any $x \in A$ has exactly $n/2$ $1$'s and hence $f$ is constant the subspace $A$ of dimension $n/2$.

Cross-post: https://mathoverflow.net/questions/41129/a-boolean-function-that-is-not-constant-on-affine-subspaces-of-large-enough-dimen
The objects you are searching for are called seedless affine dispersers with one output bit.  More generally, a seedless disperser with one output bit for a family $\mathcal{F}$ of subsets of $\{0,1\}^n$ is a function $f : \{0,1\}^n \to \{0,1\}$ such that on any subset $S \in \mathcal{F}$, the function $f$ is not constant.  Here, you are interested in $\mathcal{F}$ being the family of affine subspaces

Ben-Sasson and Kopparty in "Affine Dispersers from Subspace Polynomials" explicitly construct seedless affine dispersers for subspaces of dimension at least $6n^{4/5}$.  The full details of the disperser are a bit too complicated to describe here.  

A simpler case also discussed in the paper is when we want an affine disperser for subspaces of dimension $2n/5+10$.  Then, their construction views ${\mathbb{F}}_2^n$ as ${\mathbb{F}}_{2^n}$ and specifies the disperser to be $f(x) = Tr(x^7)$, where $Tr: {\mathbb{F}}_{2^n} \to {\mathbb{F}}_2$ denotes the trace map: $Tr(x) = \sum_{i=0}^{n-1} x^{2^i}$.  A key property of the trace map is that $Tr(x+y) = Tr(x) + Tr(y)$.