Can we design a geometry where the angle between two lines can increase infinitely? As a contrast, the maximum angle between two lines is 90° in Euclidean geometry.
I am just thinking people living in the universe with unlimited line angles will be curious that there exists a world where line angles are limited to a certain value, just like people are curious about the fact that in Minkowski geometry there is a maximum speed, i.e., the speed of light.
 A: In Special Relativity, one could define the [rapidity] angle between two lightlike directions to infinite since the area of the sector is infinite:
$$\int_0^\infty\frac{1}{x}dx\quad\mbox{diverges}$$
https://www.wolframalpha.com/input?i=integral+1%2Fx+from+0+to+inf
The [rapidity] angle between two future-timelike directions has no limit.
So, Minkowski spacetime geometry has this feature.
A: *

*The exact analogue of angle from Euclidean geometry, in Lorentz geometry, is the rapidity. Its formula is $\tanh ^{-1} \frac{v}{c}$, or $\tanh ^{-1} v$ with $c=1$


*The analogue of $v$ in Euclidean geometry is not the angle. It's the slope $m$. $v$ and $m$ literally equal the slope in the space/spacetime diagrams. Angle is related to slope by the formula $tan ^{-1} m$.


*Euclidean geometry with the metric $x^2+y^2$, is the $c=1$ version of the general Euclidean geometry . In general, the Euclidean geometry metric can be $c^2x^2+y^2$. The equivalent of rotations here would be "going in an Ellipse".


*The $c\rightarrow \infty$ limit, of both Euclidean and Lorentz geometries, is the Galilean geometry. Basically, the invariant metric of both geometries can be written as $x^2 \pm \frac{1}{c^2}y^2$. Only $x^2$ remains as the invariant quantity in the $c\rightarrow \infty$ limit, which corresponds to an "absolute $x$ co-ordinate" (like absolute time).
So, a Galilean geometry is sort of what you are looking for. But Galilean transformations do not allow for greater than 90 degree angles. Angles, defined according to $\tan ^{-1} \frac{v}{c}$, become 0 in the $c\rightarrow \infty$ limit. Your mistake was in identifying the speed limit from relativity, with the 90 degree limit from Euclidean geometry. $c$ from relativity is instead identified with the $c$ in $c^2x^2+y^2$. But this $c$ from Euclidean geometry does not represent any "speed limit".
