To name a few very public ones : tscompdata, FRED, Google Trends, Wikipedia,  Eurostat, National Center for Environmental Information, UCI Machine Learning Repository, UEA& UCR Time Series Classification Repository. Notice that I am not providing the URL. Google those names to find them.  We use datasets from these sources throughout the book. You can find them in the R Programs page (which also contains the data). 

An overview of time series that can be found in R packages is given in the CRAN Task View Time Series (if you scroll down to the time series data section). 

But each domain of expertise has its own time series data. The entertainment industry, for example, has its own propietary data. The same applies to health care, etc. 

Most time series data producers have their own API, and it is likely that R has a function to implement that API if the data producer is a government agency, or some other big established data producer.
An overview of data (some not time series, some time series) can be seen at Open Data Cran Task View. 

To see an illustration of how an API  for time serie data that we use in Chapter 1 works to access time series data from R, go to the lectures in Chapter 1, and view the lecture and video on  Quandl API. The address for te YouTube video is in Lecture Chapter 1-4 Quandl-Demo.