Hola Philip
It would sound a bit odd and maybe ambiguous if you use the infinitive because it wouldn't be very clear whether the speaker means "He asked me to get dressed" or "He asked me if he could dress me". The latter is a possible interpretation of that sentence using the infinitive because generally the use of the infinitive with pedir means that there is only one subject, not two.
Using que + subjunctive would leave no other interpretation but: He asked me to get dressed (he asked/ I get dressed), which is what it is meant to be saying, really.
Saludos
