There is only one main "coloring" agent, or pigment, in the eyes. That pigment is melanin. To a much lesser extent, at least in health individuals, you can have lipofuscin produced which gives a golden-amber color to the eyes. It should be noted that lipofuscin is most likely produced because of oxidation damage/stress in your eye, and not something to be hoping for (Beatty et al has a good review of the subject (1)).
Indeed, lipofuscin in the eyes can be seen as a symptom to physicians, so for the rest of the answer I will only consider melanin, eumelanin and pheomelanin in particular, as the normative pigment in the iris.
To make additional colors, we would need to develop an additional pigment, and or modify eumelanin so that it produces different scattering properties. So why do we have melanocytes in the eye producing melanin in the first place?
As much as we might consider it as a factor now, sexual selection was probably not the main pressure for having pigment in the eyes. Instead UV protection almost assuredly is (2, 3). Thus for another pigment to be selected for, it would probably need to do a better job protecting us vs. UV (also, further selection in humans is hampered by all kinds of things like behavior, medicine, and life span).
Eumelanin does a good job protecting us from UV (2, 3) and in fact does a better job than phenomelanin (4). It seems incredibly unlikely that we would develop the necessary evolutionary pressure to have our pigment epithelium to produce a totally foreign pigment.
And that is what prevents us from naturally having a whole rainbow of colors in our iris. One could conceivably engineer a better pigment, and have the genetic information needed to produce it delivered in gene therapy (I hate to think of people purposefully infecting their pigment epithelia). I, for one, wouldn’t sign up for that.
(1) Beatty S. et al. The role of oxidative stress in the pathogenesis of age-related macular degeneration. Surv Ophthalmol. 2000 Sep-Oct;45(2):115-34.
(2) Hu DN. et al. Role of ocular melanin in ophthalmic physiology and pathology. Photochem Photobiol. 2008 May-Jun;84(3):639-44. doi: 10.1111/j.1751-1097.2008.00316.x. Epub 2008 Mar 8.
(3) Simon JD, Peles DN. The red and the black. Acc Chem Res. 2010 Nov 16;43(11):1452-60. doi: 10.1021/ar100079y. Epub 2010 Aug 24.
(4) Hu DN et al. Uveal melanocytes, ocular pigment epithelium, and Müller cells in culture: in vitro toxicology. Int J Toxicol. 2002 Nov-Dec;21(6):465-72.