Somehow, I find your wish a bit selfcontradicting.Using a competingrisk model means exactly that persons are not in risk of any of the outcomes when the first outcome has occured. A competingrisk model is therefore not the right model analyzing recurrent events.
But, what you can do is to model the recurrent events with a Cox-model using either time since last event. You can thereby have more events per person in a Cox-model, just restart the time at each event. Or you can have time since beginning of trial as underlying time axis, then you should have the last timepoint as a left-truncation on the time at risk (you can find an example here: SAS/STAT(R) 13.2 User's Guide)
The other eventtypes can be taken into account by constructing a timedependent covariate that counts numbers of the other event type.
