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For the version of the question about finite products and coproducts:

• Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).

• Take any object $A$ in a category with all finite products and consider the full subcategory whose objects are all finite powers of $A$. This will usually fail to have coproducts. For example, the category of all finite sets whose cardinalities are powers of $2$.

For the version of the question about small products and coproducts:

• The second example generalizes easily to arbitrary powers. For example, the category of all sets of cardinality $2^\kappa$ for arbitrary cardinals $\kappa$.

• I originally claimed that the first example above generalizes easily as well, to complete meet-semilattices, but Max New correctly points out in the comments that every complete meet-semilattice is a lattice. However, as Tim Campion also correctly points out in the comments, a large semilattice may have all small meets without having all small joins. For example, the dual of the join-semilattice of all small subsets of a large set (which has no bottom element, hence no empty join).

Yet another naturalish example is the category of non-empty sets, which has all products but no initial object, hence no empty coproduct.

For the version of the question about finite products and coproducts:

• Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).

• Take any object $A$ in a category with all finite products and consider the full subcategory whose objects are all finite powers of $A$. This will usually fail to have coproducts. For example, the category of all finite sets whose cardinalities are powers of $2$.

For the version of the question about small products and coproducts:

• The second example generalizes easily to arbitrary powers. For example, the category of all sets of cardinality $2^\kappa$ for arbitrary cardinals $\kappa$.

• I originally claimed that the first example above generalizes easily as well, to complete meet-semilattices, but Max New correctly points out in the comments that every complete meet-semilattice is a lattice. However, as Tim Campion also correctly points out in the comments, a large semilattice may have all small meets without having all small joins. For example, the dual of the join-semilattice of all small subsets of a large set (which has no bottom element, hence no empty join).

Yet another naturalish example is the category of non-empty sets, which has all products but no initial object, hence no empty coproduct.

For the version of the question about finite products and coproducts:

• Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).

• Take any object $A$ in a category with all finite products and consider the full subcategory whose objects are all finite powers of $A$. This will usually fail to have coproducts. For example, the category of all finite sets whose cardinalities are powers of $2$.

For the version of the question about small products and coproducts:

• The second example generalizes easily to arbitrary powers. For example, the category of all sets of cardinality $2^\kappa$ for arbitrary cardinals $\kappa$.

• I originally claimed that the first example above generalizes easily as well, to complete meet-semilattices, but Max New correctly points out in the comments that every complete meet-semilattice is a lattice. However, as Tim Campion also correctly points out in the comments, a large semilattice may have all small meets without having all small joins. For example, the dual of the join-semilattice of all small subsets of a large set (which has no bottom element, hence no empty join).

For the version of the question about finite products and coproducts:

• Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).

• Take any object $A$ in a category with all finite products and consider the full subcategory whose objects are all finite powers of $A$. This will usually fail to have coproducts. For example, the category of all finite sets whose cardinalities are powers of $2$.

For the version of the question about small products and coproducts:

• The second example generalizes easily to arbitrary powers. For example, the category of all sets of cardinality $2^\kappa$ for arbitrary cardinals $\kappa$.

• I originally claimed that the first example above generalizes easily as well, to complete meet-semilattices, but Max New correctly points out in the comments that every complete meet-semilattice is a lattice. However, as Tim Campion also correctly points out in the comments, a large semilattice may have all small meets without having all small joins. For example, the dual of the join-semilattice of all small subsets of a large set (which has no bottom element, hence no empty join).

Yet another naturalish example is the category of non-empty sets, which has all products but no initial object, hence no empty coproduct.

• Any complete meet-semilattice that does not have joins. For example, the semilattice of subsets of $\mathbb{N}$ of size at most $n$ for a fixed $n$.
• Take any object $A$ in a category with all products and consider the full subcategory whose objects are all powers of $A$. This will usually fail to have coproducts. For example, the category of all sets whose cardinalities are powers of $2$.

For the version of the question about finite products and coproducts:

• Any meet-semilattice that does not have joins. For example, a linear order with no least element (which has no empty join).

• Take any object $A$ in a category with all finite products and consider the full subcategory whose objects are all finite powers of $A$. This will usually fail to have coproducts. For example, the category of all finite sets whose cardinalities are powers of $2$.

For the version of the question about small products and coproducts:

• The second example generalizes easily to arbitrary powers. For example, the category of all sets of cardinality $2^\kappa$ for arbitrary cardinals $\kappa$.

• I originally claimed that the first example above generalizes easily as well, to complete meet-semilattices, but Max New correctly points out in the comments that every complete meet-semilattice is a lattice. However, as Tim Campion also correctly points out in the comments, a large semilattice may have all small meets without having all small joins. For example, the dual of the join-semilattice of all small subsets of a large set (which has no bottom element, hence no empty join).