What does "smallest" exactly mean in the definition of $\mathscr B(\mathbb R)$? The only "bigger" σ-algebras I can see are the ones where we count twice, thrice, etc each open set.
It seems to me obvious that we shall not count twice, thrice etc the open sets. Ignoring these double, triple counting σ-algebras, leaves us with only one σ-algebra $-$ the basic one (the one we call smallest I think).
It means that any $\sigma$-algebra $S$ on $\Bbb R$ that contains all open sets will also contain $\mathscr{B}(\Bbb R)$.
Two examples of strictly larger algebras are the Lebesgue algebra and the trivial power set algebra (all sets are measurable), assuming the axiom of choice as normal in analysis.