The /var/log filesystem is probably filling up because some process is writing log entries over and over and over.
This can happen not only on OpenBSD but on any BSD or Linux or Unix like system.
Look at the /var directory and try to figure out what is being written:
cd /var
ls -altr
cd /var/log
ls -altr
The use something like the "tail" command to look at the end of the log files that are, for example to look at the last lines of the messages file:
cd /var/log
tail messages
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