Can I also study for a while at a university with which Leiden has no exchange agreement?

Yes, you can. But you will probably have to pay tuition fees to the university abroad. Tuition fee rates differ greatly per country and sometimes also per university. As a rule, tuition fee rates in Europe are much lower than those outside Europe, with the exception of the UK which has very high tuition fee rates. Another disadvantage is that Leiden University cannot help you enrol and you will not qualify for an Erasmus grant.