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This is a complement to Joel´s answer and some further generalizations.

In "Paracompactness and product spaces" (1948), Stone proved that if a product space is Lindelof and regular then all but countably many factors are compact.

In "Compact factors in finally compact products of topological spaces" (2005), Lipparini removed the regularity condition and generalized the result to weaker forms of compactness. For instance, it follows that if $X^{\aleph_{\alpha+n+1}}$ is finally $\aleph_{\alpha+n+1}$-compact then $X$ is finally $\aleph_\alpha$-compact (a space is finally $\kappa$-compact if any open cover admits a subcover of size less than $\kappa$). A corollary of this is that if $X^{\aleph_n}$ is finally $\aleph_n$-compact then $X$ is compact. In particular if $X^{\aleph_1}$ is Lindelof then $X$ is compact.

In a different direction (generalizing Tychonoff), in "Products of initially m-compact spaces" (1974), Stephenson and Vaughan proved that if $\kappa$ is a singular strong limit cardinal, then any product of initially $\kappa$-compact spaces is initially $\kappa$-compact (a space is initially $\kappa$-compact if any open cover of size $\kappa$ admits a finite subcover). Note that initially $\aleph_0$-compact is just countably compact, and that there are spaces $X$ such that $X$ is countably compact but $X^2$ is not (see Novák´s "On the cartesian product of two compact spaces", 1953).

All the information above was taken from: http://biblioteca.uniandes.edu.co/Tesis_2006_primer_semestre/00006522.pdf

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This is a complement to Joel´s answer and some further generalizations.

In "Paracompactness and product spaces" (1948), Stone proved that if a product space is Lindelof and regular then all but countably many factors are compact.

In "Compact factors in finally compact products of topological spaces" (2005), Lipparini removed the regularity condition and generalized the result to weaker forms of compactness. For instance, it follows that if $X^{\aleph_{\alpha+n+1}}$ is finally $\aleph_{\alpha+n+1}$-compact then $X$ is finally $\aleph_\alpha$-compact (a space is finally $\kappa$-compact if any open cover admits a subcover of size less than $\kappa$). A corollary of this is that if $X^{\aleph_n}$ is finally $\aleph_n$-compact then $X$ is compact. In particular if $X^{\aleph_1}$ is Lindelof then $X$ is compact.

In a different direction (generalizing Tychonoff), in "Products of initially m-compact spaces" (1974), Stephenson and Vaughan proved that if $\kappa$ is a singular strong limit cardinal, then any product of initially $\kappa$-compact spaces is initially $\kappa$-compact (a space is initially $\kappa$-compact if any open cover of size $\kappa$ admits a finite subcover). Note that initially $\aleph_0$-compact is just countably compact, and that there are spaces $X$ such that $X$ is countably compact but $X^2$ is not (see Novák´s "On the cartesian product of two compact spaces", 1953).