%>% has no builtin meaning but the user (or a package) is free to define operators of the form %whatever% in any way they like. For example, this function will return a string consisting of its left argument followed by a comma and space and then it's right argument.
"%,%" <- function(x, y) paste0(x, ", ", y)
# test run
"Hello" %,% "World"
## [1] "Hello, World"
The base of R provides %*% (matrix mulitiplication), %/% (integer division), %in% (is lhs a component of the rhs?), %o% (outer product) and %x% (kronecker product). It is not clear whether %% falls in this category or not but it represents modulo.
expm The R package, expm, defines a matrix power operator %^%. For an example see Matrix power in R .
operators The operators R package has defined a large number of such operators such as %!in% (for not %in%). See http://cran.r-project.org/web/packages/operators/operators.pdf
igraph This package defines %--% , %->% and %<-% to select edges.
lubridate This package defines %m+% and %m-% to add and subtract months and %--% to define an interval. igraph also defines %--% .
magrittr In the case of %>% the magrittr R package has defined it as discussed in the magrittr vignette. See http://cran.r-project.org/web/packages/magrittr/vignettes/magrittr.html
magittr has also defined a number of other such operators too. See the Additional Pipe Operators section of the prior link which discusses %T>%, %<>% and %$% and http://cran.r-project.org/web/packages/magrittr/magrittr.pdf for even more details.
dplyr The dplyr R package used to define a %.% operator which is similar; however, it has been deprecated and dplyr now recommends that users use %>% which dplyr imports from magrittr and makes available to the dplyr user. As David Arenburg has mentioned in the comments this SO question discusses the differences between it and magrittr's %>% : Differences between %.% (dplyr) and %>% (magrittr)
pipeR The R package, pipeR, defines a %>>% operator that is similar to magrittr's %>% and can be used as an alternative to it. See http://renkun.me/pipeR-tutorial/
The pipeR package also has defined a number of other such operators too. See: http://cran.r-project.org/web/packages/pipeR/pipeR.pdf
postlogic The postlogic package defined %if% and %unless% operators.
wrapr The R package, wrapr, defines a dot pipe %.>% that is an explicit version of %>% in that it does not do implicit insertion of arguments but only substitutes explicit uses of dot on the right hand side. This can be considered as another alternative to %>%. See https://winvector.github.io/wrapr/articles/dot_pipe.html
Bizarro pipe. This is not really a pipe but rather some clever base syntax to work in a way similar to pipes without actually using pipes. It is discussed in http://www.win-vector.com/blog/2017/01/using-the-bizarro-pipe-to-debug-magrittr-pipelines-in-r/ The idea is that instead of writing:
1:8 %>% sum %>% sqrt
## [1] 6
one writes the following. In this case we explicitly use dot rather than eliding the dot argument and end each component of the pipeline with an assignment to the variable whose name is dot (.) . We follow that with a semicolon.
1:8 ->.; sum(.) ->.; sqrt(.)
## [1] 6
Update Added info on expm package and simplified example at top. Added postlogic package.