A Boolean function that is not constant on affine subspaces of large enough dimension
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 $Tr(x+y) = Tr(x) + Tr(y)$.