How many dendrite connections vs axon terminals does a multipolar cerebral neuron have? I find countless places stating that neurons have tens of thousands of "connections" or "synapses" and one axon.

Do neurons have tens of thousands of dendrite connections, and a few axon terminals; or a few dendrite connections and tens of thousands of axon terminals; or tens of thousands of each?

When a neuron's action potential is triggered, how many other neurons does it affect?
Multipolar neurons have multiple inputs (dendritic connections), and one output (the axon).There are also bipolar cells in the retina, these have one dendrite (input) and one axon (output) (Nelson &amp; Connaughton, 2012).

Regarding dendrites:
A striking example are the Purkinje cells in the cortex. These cells have elaborate dendritic trees making 100,000 to 200,000 connections, but still there is just one axon. Hence they integrate massive amounts of (sensory) information and funnel it into one output signal (Purves et al., 2002).

Hence, dependent on the cell type, neurons can have one or as many as 200k dendritic connections. 

Regarding axon terminals:
As far as I am aware, all neurons have just one axon. The axon can target neurons along the way (en passant) and the axon can terminate in multiple terminals contacting various cells. Some neurons contain one terminal (e.g. bipolar cells), others as many as thousands of terminals (Brady et al., 2012).

The only situation where multiple axons arise from one cell is when the axon bifurcates along the way, sending one or more collaterals from the axon off back to the cell. Basically, the neuron has still just one axonal output, but collateral regulatory info is sent off back to the cell.

There are situations where multiple axons arise, but that occurs only in neurons that have been tinkered around with genetically.

References
- Brady et al., Basic Neurochemistry. 8th ed (2012)
- Nelson &amp; Connaughton, Bipolar Cell Pathways in the Vertebrate Retina. In: Webvision (2012)
- Purves et al. (eds), Neuroscience. 2nd ed. Sunderland (MA): Sinauer Associates