Sometimes these algebras are fiber products along surjective homomorphisms; for example $k[x,xy,xy^2,\dotsc] \cong k[x,y] \times_{k[y]} k$homomorphisms, hence the spectrum is $\mathbb{A}^2_k$ with the line $x=0$ contracted to a point. This is example 3.5 inand Karl Schwede's paper on Gluing Schemes. Another example is tells us how the subalgebra $\{(a,b) \in \mathbb{Z}^2 : a \equiv b \text{ mod } 6\}$. The spectrum is the gluing of two copies of $\mathrm{Spec}(\mathbb{Z})$ along the closed points $(2),(3)$. Interestingly, if you localize at all primes $\neq 2,3$, you will get an affine scheme which is weakly homotopy equivalent to $S^1$, the Pseudocirclelooks like. For example:
The spectrum of $k[x,xy,xy^2,\dotsc] \cong k[x,y] \times_{k[y]} k$ is $\mathbb{A}^2_k$ with the line $x=0$ contracted to a point. This is example 3.5 in the cited paper.
The spectrum of $k[x^2,x^3] \cong k[x] \times_{k[x]/(x^2)} k$ is $\mathbb{A}^1_k$ with a rational point doubled, i.e. the cuspidal curve.
The spectrum of $k[x^2-1,x^3-x] \cong k[x] \times_{k[x]/((x-1)(x+1))} k$ is $\mathbb{A}^1_k$ with two rational points identified, i.e. the nodal curve.
The spectrum of $\{(a,b) \in \mathbb{Z}^2 : a \equiv b \text{ mod } 6\}$ is the gluing of two copies of $\mathrm{Spec}(\mathbb{Z})$ along the closed points $(2),(3)$. Interestingly, if you localize at all primes $\neq 2,3$, you will get an affine scheme which is weakly homotopy equivalent to $S^1$, the Pseudocircle.