TITLE: Examples of plane algebraic curves
QUESTION [12 upvotes]: There are many interesting sequences of polynomials which contain
polynomials of arbitrarily high degree, for example classical
orthogonal polynomials. Most of them arise as characteristic polynomials
of some sequences of operators, or as polynomial solutions
of some differential equations.

What are some natural specific
sequences of plane (affine or projective) algebraic
curves which contain curves of arbitrarily high degree and genus?

One such example is Fermat's curves $x^n+y^n=1$. Lissajous
(a.k.a. Chebyshev)
curves are of arbitrary degree but they have zero genus.  Sequences of hyperelliptic curves occur in the theory of integrable systems. What else?
I looked to the Catalog of Plane curves by D. Lawrence (Dover, 2014)
and to the book of Brieskorn and Knörrer, Plane algebraic curves,
and found only Lissajous curves, epitrochoids and
hypotrochoids (all of genus zero) as examples of arbitrarily high degree.
I understand that many examples can be constructed. But I am asking on some naturally occuring sequences, whatever it can mean. Of some historical significance or appearing in applications.
EDIT. Thanks to all who answered or commented. I am not marking this question as "answered" for a while, hoping for more examples. Of course, <a href="https://en.wikipedia.org/wiki/Classical_modular_curve>classical modular curves belong here, thanks to Felipe Voloch.
Let me mention my motivation for this question. For some time I am studying what can be called "Lamé modular curves" (surprisingly, there is no established name for them). Lamé functions
are solutions of Lamé's differential equation whose squares are polynomials. Existence of
such a solution imposes a polynomial equation connecting the modulus of the
torus $J$ and an "accessory parameter". These polynomials define a family of plane affine algebraic curves
which contains curves of arbitrary degree and genus, and their coefficients are integers.

REPLY [3 votes]: What you call Fermat curves, are also referred to as Lamé curves, since Lamé proposed these as tools for physics, in particular crystals.  I started using them in the 1990s to describe the shapes of plants.  Recently we tested this on many botanical samples, see e.g.

Huang, Weiwei; Li, Yueyi; Niklas, Karl J.; Gielis, Johan; Ding, Yongyan; Cao, Li; Shi, Peijian, A Superellipse with Deformation and Its Application in Describing the Cross-Sectional Shapes of a Square Bamboo, Symmetry 12 no. 12 (2020) 2073 https://doi.org/10.3390/sym12122073

Yours,
Johan