Is it possible to host interactive R Markdown files on Github Pages?

When you render an rmd file you can also knit it to HTML as well and you can then host that page on github pages.

While it's not possible to host fully-fledged Shiny apps on Github pages (Indeed, as @Gregor suggested, shinyapps.io is useful for this), the devs for Shiny have been working to make some of the functionality run completely on the client-side via htmlwidgets.

Here is a simple example running on Github pages:

README.Rmd

## Example of displaying htmlwidgets on a Github pages site

```{r}
# Source: http://www.htmlwidgets.org/showcase_plotly.html
library(plotly)
p <- ggplot(data = diamonds, aes(x = cut, fill = clarity)) +
            geom_bar(position = "dodge")
ggplotly(p)
```

Rendered HTML

(Live version: Github pages htmlwidget demo)

For more complex interactions, including communicating between widgets entirely on the client-side, check out Joe Cheng's recent crosstalk demo from UserR! 2016.