This should really be a comment on the answer of Steven Sam, but it doesn't fit in the box. The systematic way to reduce to the cases described by Sam is via castling transformations.

It is shown in a paper of P.G. Parfenov that the group $G = GL(k_{1}, \mathbb{C})\times \dots \times GL(k_{r}, \mathbb{C})$ has finitely many orbits in $V = \mathbb{C}^{k_{1}}\otimes \dots \otimes \mathbb{C}^{k_{r}}$ if and only if $(k_{1}, \dots, k_{r})$ is one of $(n)$, $(m, n)$, $(2, 2, n)$, or $(2, 3, n)$, where $n \geq 3$. This is published in:

http://iopscience.iop.org/1064-5616/192/1/A05

(This also follows from Theorem $2$ of V.G. Kac's paper "Some remarks on nilpotent orbits")

That $G$ have finitely many orbits on $V$ means that $(G, V)$ is prehomogeneous ($G$ has a Zariski dense orbit), but the converse is not true. One can ask also when $(G, V)$ is prehomogeneous. A necessary condition was stated above by R. Bryant in a comment to the original question (for $r > 3$ the condition becomes $\sum_{i}k_{i}^{2} - \prod_{i}k_{i} \geq r- 1$). In general this can be resolved by using the Sato-Kimura classification of reduced irreducible prehomogeneous vector spaces. Here reduced means reduced with respect to castling equivalence, which is a notion of equivalence of prehomogeneous vector spaces based on the duality of Grassmannians (see Sato-Kimura and Kimura's book on prehomogeneous vector spaces). One can show that every $(G, V)$ that is prehomogeneous is castling equivalent to such a space with finitely many orbits, that is in the list due to Parfenov mentioned above (I have not been able to find this statement as such published anywhere, but it is known). For general formats $(k_{1}, \dots, k_{r})$ I don't know how to state this cleanly simply as a condition on the $k_{i}$'s, but in practice it is straightforward to check (the necessary details can be gleaned from section 3 of Sato-Kimura). For example, if $r = 3$ then the dimensions have the form $(a, b, c)$. If $ab > c$ then this format is castling equivalent to $(a, b, c - ab)$, and via this remark, one can reduce to a castling equivalent format in which (after reordering) $c \leq 2$.

In some cases the fundamental relative invariant is a hyperdeterminant. This occurs, for example, for the formats $(2, 2, 2)$ (the classical Cayley hyperdeterminant) and $(2, 3, 3)$. See chapter 14 of the book of Gelfand, Kapranov, and Zelevinsky.

This should really be a comment on the answer of Steven Sam, but it doesn't fit in the box. The systematic way to reduce to the cases described by Sam is via castling transformations.

It is shown in a paper of P.G. Parfenov that the group $\DeclareMathOperator\GL{GL}G = \GL(k_{1}, \mathbb{C})\times \dots \times \GL(k_{r}, \mathbb{C})$ has finitely many orbits in $V = \mathbb{C}^{k_{1}}\otimes \dots \otimes \mathbb{C}^{k_{r}}$ if and only if $(k_{1}, \dots, k_{r})$ is one of $(n)$, $(m, n)$, $(2, 2, n)$, or $(2, 3, n)$, where $n \geq 3$. This is published in: Orbits and their closures in the spaces $\mathbb C^{k_1} \otimes \dotsb \otimes \mathbb C^{k_r}$.

(This also follows from Theorem 2 of V.G. Kac's paper "Some remarks on nilpotent orbits".)

That $G$ have finitely many orbits on $V$ means that $(G, V)$ is prehomogeneous ($G$ has a Zariski dense orbit), but the converse is not true. One can ask also when $(G, V)$ is prehomogeneous. A necessary condition was stated above by R. Bryant in a comment to the original question (for $r > 3$ the condition becomes $\sum_{i}k_{i}^{2} - \prod_{i}k_{i} \geq r- 1$). In general this can be resolved by using the Sato–Kimura classification of reduced irreducible prehomogeneous vector spaces. Here reduced means reduced with respect to castling equivalence, which is a notion of equivalence of prehomogeneous vector spaces based on the duality of Grassmannians (see Sato-Kimura and Kimura's book on prehomogeneous vector spaces). One can show that every $(G, V)$ that is prehomogeneous is castling equivalent to such a space with finitely many orbits, that is in the list due to Parfenov mentioned above (I have not been able to find this statement as such published anywhere, but it is known). For general formats $(k_{1}, \dots, k_{r})$ I don't know how to state this cleanly simply as a condition on the $k_{i}$'s, but in practice it is straightforward to check (the necessary details can be gleaned from section 3 of Sato-Kimura). For example, if $r = 3$ then the dimensions have the form $(a, b, c)$. If $ab > c$ then this format is castling equivalent to $(a, b, c - ab)$, and via this remark, one can reduce to a castling equivalent format in which (after reordering) $c \leq 2$.

In some cases the fundamental relative invariant is a hyperdeterminant. This occurs, for example, for the formats $(2, 2, 2)$ (the classical Cayley hyperdeterminant) and $(2, 3, 3)$. See chapter 14 of the book of Gelfand, Kapranov, and Zelevinsky.

This should really be a comment on the answer of Steven Sam, but it doesn't fit in the box. The systematic way to reduce to the cases described by Sam is via castling transformations.

It is a theorem of P.G. Parfenov that the group $G = GL(k_{1}, \mathbb{C})\times \dots \times GL(k_{r}, \mathbb{C})$ has finitely many orbits in $V = \mathbb{C}^{k_{1}}\otimes \dots \otimes \mathbb{C}^{k_{r}}$ if and only if $(k_{1}, \dots, k_{r})$ is one of $(n)$, $(m, n)$, $(2, 2, n)$, or $(2, 3, n)$, where $n \geq 3$. This is published in:

http://iopscience.iop.org/1064-5616/192/1/A05

That $G$ have finitely many orbits on $V$ means that $(G, V)$ is prehomogeneous ($G$ has a Zariski dense orbit), but the converse is not true. One can ask also when $(G, V)$ is prehomogeneous. A necessary condition was stated above by R. Bryant in a comment to the original question (for $r > 3$ the condition becomes $\sum_{i}k_{i}^{2} - \prod_{i}k_{i} \geq r- 1$). In general this can be resolved by using the Sato-Kimura classification of reduced irreducible prehomogeneous vector spaces. Here reduced means reduced with respect to castling equivalence, which is a notion of equivalence of prehomogeneous vector spaces based on the duality of Grassmannians (see Sato-Kimura and Kimura's book on prehomogeneous vector spaces). One can show that every $(G, V)$ that is prehomogeneous is castling equivalent to such a space with finitely many orbits, that is in the list due to Parfenov mentioned above (I have not been able to find this statement as such published anywhere, but it is known). For general formats $(k_{1}, \dots, k_{r})$ I don't know how to state this cleanly simply as a condition on the $k_{i}$'s, but in practice it is straightforward to check (the necessary details can be gleaned from section 3 of Sato-Kimura). For example, if $r = 3$ then the dimensions have the form $(a, b, c)$. If $ab > c$ then this format is castling equivalent to $(a, b, c - ab)$, and via this remark, one can reduce to a castling equivalent format in which (after reordering) $c \leq 2$.

In some cases the fundamental relative invariant is a hyperdeterminant. This occurs, for example, for the formats $(2, 2, 2)$ (the classical Cayley hyperdeterminant) and $(2, 3, 3)$. See chapter 14 of the book of Gelfand, Kapranov, and Zelevinsky.

This should really be a comment on the answer of Steven Sam, but it doesn't fit in the box. The systematic way to reduce to the cases described by Sam is via castling transformations.

It is shown in a paper of P.G. Parfenov that the group $G = GL(k_{1}, \mathbb{C})\times \dots \times GL(k_{r}, \mathbb{C})$ has finitely many orbits in $V = \mathbb{C}^{k_{1}}\otimes \dots \otimes \mathbb{C}^{k_{r}}$ if and only if $(k_{1}, \dots, k_{r})$ is one of $(n)$, $(m, n)$, $(2, 2, n)$, or $(2, 3, n)$, where $n \geq 3$. This is published in:

http://iopscience.iop.org/1064-5616/192/1/A05

(This also follows from Theorem $2$ of V.G. Kac's paper "Some remarks on nilpotent orbits")

That $G$ have finitely many orbits on $V$ means that $(G, V)$ is prehomogeneous ($G$ has a Zariski dense orbit), but the converse is not true. One can ask also when $(G, V)$ is prehomogeneous. A necessary condition was stated above by R. Bryant in a comment to the original question (for $r > 3$ the condition becomes $\sum_{i}k_{i}^{2} - \prod_{i}k_{i} \geq r- 1$). In general this can be resolved by using the Sato-Kimura classification of reduced irreducible prehomogeneous vector spaces. Here reduced means reduced with respect to castling equivalence, which is a notion of equivalence of prehomogeneous vector spaces based on the duality of Grassmannians (see Sato-Kimura and Kimura's book on prehomogeneous vector spaces). One can show that every $(G, V)$ that is prehomogeneous is castling equivalent to such a space with finitely many orbits, that is in the list due to Parfenov mentioned above (I have not been able to find this statement as such published anywhere, but it is known). For general formats $(k_{1}, \dots, k_{r})$ I don't know how to state this cleanly simply as a condition on the $k_{i}$'s, but in practice it is straightforward to check (the necessary details can be gleaned from section 3 of Sato-Kimura). For example, if $r = 3$ then the dimensions have the form $(a, b, c)$. If $ab > c$ then this format is castling equivalent to $(a, b, c - ab)$, and via this remark, one can reduce to a castling equivalent format in which (after reordering) $c \leq 2$.

In some cases the fundamental relative invariant is a hyperdeterminant. This occurs, for example, for the formats $(2, 2, 2)$ (the classical Cayley hyperdeterminant) and $(2, 3, 3)$. See chapter 14 of the book of Gelfand, Kapranov, and Zelevinsky.