It depends on contracted power (maximum power the electricity provider will supply you as per your contract with them)

Were I am (Portugal) the typical contracted power for an appartment building is 3.45kW, so just slightly above those 3000W you mentioned, though you can choose a higher one such as 6.9kW (note that your circuit breaker should be sized for it).

The default and available values for retail customers probably vary from country to country, but given that even a cheap microwave oven uses 700W, an electric kettle about 1000W and an electric heater even more than that, I would say that anything below 1000W is unlikely (and judging by your oven power usage, it's probably more than 3kW in at least most places).

Could be that the site I had wasn't accurate. If so, that just means the prospect of charging it using portable solar panels is even more out of reach.

A toaster is 1 kW.