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This example generalizes some of the others already mentioned: Take an infinite family of vector spaces $(V_i)_{i \in I}$. Now what about $\prod_{i \in I} V_i$, can you write down a basis?
Also, it is easy to construct an infinite multilinear tensor product $\bigotimes_{i \in I} V_i$. However, writing down a basis is equivalent to find a set of representatives of $\prod_{i \in I} V_i \setminus \{0\} / \sim$, where $\sim$ identifies families of elements, which differ only at finitely many indices. And this cannot be done explicitely.