What determines the number of diffraction maxima that occur? I recently performed an experiment using an He Ne laser and was observing Fourier transformations of the laser light using lenses.  After using a grid to diffract the laser light, (50 lines/mm) a diffraction pattern was observed. 
However, I asked my lab demonstrator 'What determines the number of diffraction maxima we can observe?'  and I was left without an answer.  From my understanding the diffraction pattern is infinite, yet a finite pattern is observed on the screen.
If I have left out any key info please let me know.
 A: There are several real world effects limiting the number of maxima observed, off the top of my head I can think of the following. For illustration I will use the double slit instead of a grating, since the fundamental concerns are the same, and reasoning about double slits is easier.
Finite width of the slits
The lines in the grating are not infinitesimally thin (if they were only an infinitesimal amount of radiation would be transmitted). The effect of this is easy to calculate in the far field for the double slit, where the result is, that the intensity after the slit is (see Wikipedia for the derivation):
\begin{align*}
 I(\theta) &= I_0 \cos^2(x) \frac{\sin(y)}{y}, & x &:= \frac{\pi d \theta}{\lambda}, & y := \frac{\pi b \theta}{\lambda}.
\end{align*}
Where $b$ is the width of the lines and $d$ is the distance between the middle of the lines. $\lambda$ is the wavelength of the incident light and $\theta$ is the angle at which we observe the pattern.
You may note, that the form of the expression is the diffraction pattern of a single slit ($\propto \sin(y)/y$) multiplied by the pattern of the double slit ($\propto \cos^2$), this rule, that the total interference pattern is the product holds for the grating as well. So by analogy, the intensity of the interference pattern will be cut off $\propto \sin(y)/y$ for the grating as well.  (Note that there is usually not a sharp cut-off after $n$ maxima, but a gradual decline in intensity).
Geometry
As the equations above show, the interference maxima are spaced evenly in the angle $\theta$, if we go to $x \to \infty$ on an even screen, the angle will not increase past $\pi/2$, therefore, the number of maxima you can observe on the screen is limited. (This can be, by the way, understood as the classical version of the momentum conservation of the photon mentioned by Jon Custer in a comment). If we have a cylindrical screen obviously the same restriction of $\theta \in (-\pi/2, \pi/2)$ applies.
Apertures after the grating
If there is an aperture in the beamline after the grating (which may be there depending on the setup), then this aperture geometrically limits the number of maxima that can be observed (by simply cutting some off).
