What are some examples of third, fourth, or fifth order logic sentences?

The axioms of topology, for example, can be seen as third-order axioms. Simply because of the axiom that a topology is closed under unions:

$$\forall\mathcal U((\forall U\in\mathcal U\rightarrow U\in\tau)\rightarrow(\exists V\forall x(x\in V\leftrightarrow\exists U\in\mathcal U(x\in U))\land V\in\tau))$$

In the language of arithmetic, a well-order of the second-order predicates (namely, $\mathcal P(\Bbb N)$), or even the existence thereof, is a third-order sentence coming from the numbers themselves.

To some extent this is the great thing about set theory here. It allows us to take any of these high-order sentences and make them first-order in the language of sets. Of course we can make them into first-order in a two/three/four-sorted logic, which acts a bit like type theory, but you do run into issues there (for example, the characterization of $\Bbb R$ as the unique complete ordered field won't translate well into first-order logic).

In the context of higher-order arithmetic, there are many natural third-order statements. In arithmetic, quantifiers over natural numbers are first-order, quantifiers over sets of natural numbers are second-order, and quantification over sets of sets of natural numbers is third-order.

Using standard coding methods, quantifying over real numbers is second-order, so quantifying over sets of real numbers is third-order.

Some English sentences that are expressed as third-order statements in the language of arithmetic, but not as second-order statements, include:

• There is an nonprincipal ultrafilter on $\mathbb{N}$.

• Every subset of the unit interval $[0,1]$ has a cluster point.

• There is a discontinuous function from $\mathbb{R}$ to $\mathbb{R}$.

Similarly, one can obtain fourth-order statements by quantifying over arbitrary subsets of $\mathbb{R}$.