Like explained in this passage that a period is a complex number whose real and imaginary parts are integrations of rational functions over $\mathbb{Q}$ on some $\mathbb{Q}$-semi-algebra set in $\mathbb R^n$. Or equivalently, or $X$ is a smooth variety over $\mathbb{Q}$ with a normal crossing divisor $D$. Periods are the matrix coefficients of the period isomorphisms for all $i$ $$H_{dR}^i(X,D)\otimes\mathbb C\to H_{sing}^i(X,D)\otimes\mathbb C.$$
Can we actually get all periods without using this normal crossing divisor, i.e., just using all period isomorphisms
$$H_{dR}^i(X)\otimes\mathbb C\to H_{sing}^i(X)\otimes\mathbb C.$$
My intuition is we can not. Although we could still generated $\pi$ by $\mathbf{G}_m$ and the unit circle. However, it seems impossible to generate $\log(n)$ where we need the segment $[1,n]$ (a non-closed chain), whereas in the second one we can only use closed chains.
