Wait, no polygons? Instalove, for me! 3 I'm OBSESSED with alternatives to polygons.
There are plenty of artistic reasons not to use NURBS etc though as Laa-Yosh went into in detail.
Yeah, personally i experimented with bezier patches for characters before the year 2000. I was pretty excited when NV even added HW acceleration for this with GeForce3 - this was fast! But they abandoned the feature with the next generation because nobody used it.
Some time later i realized it's indeed pointless. For characters Catmull Clark subdivision is so much better. Modeling with bezier patches is true pain.
Recently there was one guy at the gamedev forum who worked on Nurbs for games. (Nurbs is basically stitched bezier patches, but for continuous curvature we need an extended form with weighted control points, called rational bezier patch IIRC.)
He listed several advantages for Nurbs:
Can be implemented efficiently with HW tessellation. (Catmull Clark is difficult, although some recent CoD game did it for characters.)
Modeling is much faster (but only for experienced CAD people, not game artists).
UV Parametrization is no extra work.
Dynamic LOD. (I said this can only increase detail, but does not help with the harder and more important problem of reducing them. He replied there would be options to remove patches, closing holes etc, but i think that's to complex to be practical.)
Less data and RAM.
Unlike Catmull Clark, NURBS are C2 continuous across irregular vertices.
But other people at the forum did not agree. Counter arguments:
Complexity of implementation.
Not artist friendly.
Personally i'm not convinced either because it's just another system to handle a single special case (likely human made stuff like vehicles or buildings. Nobody would want to model and animate Nurbs characters).
It sounds more reasonable to convert your Nurbs models to triangle meshes and work on a general LOD system for this instead. But i also think it's good some people still explore those alternatives...
However, parametric geometry is technically not an alternative to triangles (like voxels, points or SDF). Trinagles are still required, and any 'downsides' of triangles are amplified.