The standard textbook neuron receives input at its dendritic tree, fires action potentials and has a single, long output axon to send the message off to distant locations.
However, non-spiking, anaxonic neurons are not a rarity; in various sensory systems there are bipolar cells present that lack a clearly defined axon and are non-spiking, such as the amacrine cells and the horizontal cells in the vertebrate retina (Fig. 1.) and the granule cells in the mammalian olfactory bulb (Hisada, 1989).
Generally, non-spiking neurons work as analogue integrators. Where a spiking neuron can only transmit a spike, or no spike (binary signal), a non-spiking neuron can convey an infinitesimal number of membrane voltages instead. Hence, it is well suited to integrate the input from various other neurons. Amacrine cells and horizontal cells lack clearly defined axonic elements and are thus non-spiking anaxonic neurons. Their dendrites, however, do form synapses with other neurons, generally on their dendrites too, forming dendro-dendritic connections. The function of the bipolar cells in the vertebrate retina are unclear.
There are other non-spiking anaxonic neurons identified. A notable example are the interneurons seen in the ganglia of the motor systems of various arthropods, including crayfish, locusts and cockroaches. Here, they are coupled to neurons that convey descending output to motoneurons, and they are thought to be involved in selecting the adequate information to be gated to the motoneurons (Takahata et al., 1981), or to be necessary for generating oscillations in the system (Pearson & Fourtner, 1975). But again, their exact functions are unclear. One reason why may be because they are local circuit neurons. In other words, they do not send their signals far off to distant locations because they lack clear axons. For example, the role of the retinal ganglion cells becomes readily apparent when you sever their long axons that are bundled in the optic nerve (one becomes blind in that eye).
Fig. 1. Retinal circuitry showing the non-spiking, anaxonic horizontal and bipolar cells. Source: David Hubel's Eye, Brain, and Vision