If you do not bother to train your neural net, what happens? In the infinite-width limit you get a Gaussian process. There are a number of net architectures which do not make use of that argument and which are still random though.
Recurrent: Echo State Machines / Random reservoir networks
This sounds deliciously lazy; At a glance it sounds like the process is to construct a random recurrent network, i.e. a network of random saturating IIR filters. Let the network converge to a steady state for a given stimulus. These are the features to which you fit your classifier/regressor/etc.
Easy to implement, that. I wonder when it actually works, constraints on topology etc.
Some of the literature claims these are be based on spiking (i.e. even-driven) models, but AFAICT this is not necessary, although it might be convenient for convergence.
Various claims are made about how hard they avoid the training difficulty of similarly basic RNNs by being essentially untrained; you use them as a feature factory for another supervised output algorithm.
Suggestive parallel with random projections. Not strictly recurrent, but same general idea: He, Wang, and Hopcroft (2016).
Lukoševičius and Jaeger (2009) maps out various types, as much as that is possible in the shifting buzzword sand of neural network research.
From a dynamical systems perspective, there are two main classes of RNNs. Models from the first class are characterized by an energy-minimizing stochastic dynamics and symmetric connections. The best known instantiations are Hopfield networks, Boltzmann machines, and the recently emerging Deep Belief Networks. These networks are mostly trained in some unsupervised learning scheme. Typical targeted network functionalities in this field are associative memories, data compression, the unsupervised modeling of data distributions, and static pattern classification, where the model is run for multiple time steps per single input instance to reach some type of convergence or equilibrium (but see e.g., Taylor, Hinton, and Roweis (2006) for extension to temporal data). The mathematical background is rooted in statistical physics. In contrast, the second big class of RNN models typically features a deterministic update dynamics and directed connections. Systems from this class implement nonlinear filters, which transform an input time series into an output time series. The mathematical background is nonlinear dynamical systems. The standard training mode is supervised.
Non-random reservoir computing
See Reservoir computing.