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An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via $$ m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\} $$ and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

Later - in a comment below Simon Henry revealed a misconception of mine: the specific examples I proposed were actually wrong. While still not understanding well what goes on, but inspired by his further comment, I decided to add some (hopefully) valid examples, with the aid of "Remarks on quintessential and persistent localizations" by Johnstone (TAC 2 (1996) pp. 90–99). It is shown there that for $M$-sets, those local operators $j$ on $\Omega$ for which the associated sheaf functor is not only left but also right adjoint to the inclusion of $j$-sheaves, form a lattice isomorphic to the lattice of central idempotents of $M$.

Explicitly, if $e$ is a central idempotent of $M$ then $\mathfrak a\mapsto e\mathfrak a$ is a local operator, with sheaves those $M$-sets on which $e$ acts by identity, the associated sheaf of an $M$-set $X$ being $eX$.

An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via $$ m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\} $$ and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

Later - in a comment below Simon revealed a misconception of mine: the actual examples I proposed were actually wrong. While still not understanding well what goes on, but inspired by his further comment, I decided to add some (hopefully) valid examples, with the aid of "Remarks on quintessential and persistent localizations" by Johnstone (TAC 2 (1996) pp. 90–99). It is shown there that for $M$-sets, those local operators $j$ on $\Omega$ for which the associated sheaf functor is not only left but also right adjoint to the inclusion of $j$-sheaves, are in one-to-one correspondence with central idempotents of $M$.

Explicitly, if $e$ is a central idempotent of $M$ then $\mathfrak a\mapsto e\mathfrak a$ is a local operator, with sheaves those $M$-sets on which $e$ acts by identity, the associated sheaf of an $M$-set $X$ being $eX$.

An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via $$ m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\} $$ and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

Caveat - Simon Henry explains in the comment below why the specific examples I propose in the end are actually wrong. Currently trying to come up with some correct examples.

An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via $$ m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\} $$ and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

Later - in a comment below Simon Henry revealed a misconception of mine: the specific examples I proposed were actually wrong. While still not understanding well what goes on, but inspired by his further comment, I decided to add some (hopefully) valid examples, with the aid of "Remarks on quintessential and persistent localizations" by Johnstone (TAC 2 (1996) pp. 90–99). It is shown there that for $M$-sets, those local operators $j$ on $\Omega$ for which the associated sheaf functor is not only left but also right adjoint to the inclusion of $j$-sheaves, form a lattice isomorphic to the lattice of central idempotents of $M$.

Explicitly, if $e$ is a central idempotent of $M$ then $\mathfrak a\mapsto e\mathfrak a$ is a local operator, with sheaves those $M$-sets on which $e$ acts by identity, the associated sheaf of an $M$-set $X$ being $eX$.

An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via $$ m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\} $$ and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via $$ m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\} $$ and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

Caveat - Simon Henry explains in the comment below why the specific examples I propose in the end are actually wrong. Currently trying to come up with some correct examples.