As an example, take a country where real GDP (measured in base year dollars) increases every year, but where the real GDP per capita (also measured in base year dollars) fluctuates, increasing and decreasing in equal measure. In this example, does the country experience economic growth every year? That is, does economic growth reflect the consistent increasing of RGDP or does it reflect the sometimes increasing, sometimes decreasing RGDP per capita?
Usually GDP growth rates are publicly talked about in real terms but not per capita, see for example World Bank where, above the relevant table we read
Annual percentage growth rate of GDP at market prices based on constant local currency. Aggregates are based on constant 2005 U.S. dollars. GDP is the sum of gross value added by all resident producers in the economy plus any product taxes and minus any subsidies not included in the value of the products. It is calculated without making deductions for depreciation of fabricated assets or for depletion and degradation of natural resources.
Real GDP levels are also used in empirical research papers and books -for example, in Blanchard & Fischer, the first figure in the book is real GNP (not per capita).
A technical issue regarding per capita magnitudes is that population is not measured quarterly, or yearly (in fact it is usually measured per decade). So any attempt to provide quarterly or yearly "GDP per capita" figures would require extrapolating the population from sparser data. While this is not necessarily catastrophic as regards accuracy, it requires more resources to be devoted to its calculation.
Per capita concepts certainly appear more often in theoretical research papers. The main reason is that theoretical models look also for possible "steady-states" (as attractors of the economy's evolution). Since GDP in levels exhibits a long-run upward trend, the next step is to consider "GDP per capita" (to "take out" the effect of population growth), and then the next step is to consider efficiency-indexed growth, like "GDP per labor efficiency unit", in order to look for the model's steady-state.
Real GDP (or even nominal GDP) is probably the most often used measurement of growth.
Can you give an example of a country where real GDP constantly grows, but real GDP per capita fluctuates between positive and negative growth? This seems, at first glance, like a very theoretical idea as it would require large fluctuations in population growth which is usually quite constant.
The recent migrations might be a special case where you might find something like this happening in the short run, but the effect will probably vary by country with different statistical methods (for example, there are differences between countries about when an asylum seeker enters the population).
As a side note, GDP per capita is a measure of the (average) standard of living. If GDP per capita increases then people (on average) are better off.
I keeps saying 'on average' because it is possible (and seems to be actually happening) that some people see no improvement in their standard of living while others benefit greatly. But note that in this situation -- at least theoretically -- government tax-and-transfer could make everyone better off.