It depends on the country. In Canada, most certainly do, as it is the only way to be promoted to Corporal after 4 years, by going from a BE, to an IE (Basic Engagement to Intermediate Engagement). Many will quite while they are on their IE. If they don't re-enlist, they remain at the private level for their full 5 years. I know one guy who actually did that - but, he was the only one.
Other countries are different. E.g., in the USA, is is very common to do the initial engagement, and that is it. Rick, above, gave the numbers which I assume are from the US.