Upper bound on the size of a Concept Lattice (Galois Lattice)?
As told in the previous comments, $min\{2^{|O|}, 2^{|A|}\}$ is a correct upper bound. 

When the parameter $R$ is also available, we can improve the upper bound to $min\{2^{|O|}, 2^{|A|}, 2^{1+\sqrt{|R|}}\}$