Modular forms of the same weight $k$ and levels $m | n$ naturally embed: $$M_k(\Gamma(m)) \rightarrow M_k(\Gamma(n)),$$ and any element of the union $\bigcup M_k(\Gamma(n))$ is an actual modular form.
There are also surjective averaging maps $$M_k(\Gamma(n)) \rightarrow M_k(\Gamma(m))$$ in the other direction. (Another way to see this is that $M_k(\Gamma(n))$ is naturally isomorphic to its dual through the Petersson scalar product, which flips the direction.) We can take the projective limit $$\varprojlim M_k(\Gamma(n)).$$ I believe this is in some sense larger than $\bigcup M_k(\Gamma(n)).$
Does such an object have geometric meaning? I wonder how the set $\varprojlim M_k(\Gamma(n))$ may be related to elliptic curves.