What is the increased privacy risk in chrome from allowing third party cookies by default

Allowing cookies is not so threatening. It cannot access your hard drive or pass on virus. But it can store some of the data and access data to certain limits.

For a lay-man to understand, cookies will be bit of information which the website stores for future access.

The modern web browser is not restricting the third party cookies by default. This is basically done to track and identify the visitor. It is not only chrome which allows, it is also firefox and IE which allows too.

You will get more advertisement while you are visiting the website.

You will be tracked or your profile will be tracked

It can track browser history.

It might create nuisance by displaying unwanted advertisement which consumes data.

It's better to allow cookies after it prompts you and confirmed by you.