Can there be an infinite amount of output neurones in a neural network?

Am I right into thinking that one output neurone can only equal one thing like dog or the number 7 for example. If yes, then how will I be able to have an infinite amount of output neurones in my neural network.

The meaning of the output of a neuron is defined by you (the developer) as well as the use case.
In a common multi-class classification scenario, each neuron will represent the logits of the corresponding class. I.e. to predict 10 classes the last linear layer would have 10 outputs (neurons).
However, you could also claim that a single output could work and see it as a regression case (I don’t think it would perform well, but in the end nothing stops you from experimenting with different approaches).

You won’t be able to use an infinite amount of neurons, as you cannot store an infinite amount of data on any system so your idea sounds like a theoretical experiment.

Thank you for your help.