Can there be any limit to maximum row of table in DBMS (specifically MySQL)?

I would like create table to save logfile and it is row increase so quick I would like understand what shoud I actually do to avoid any difficulty.

You need to periodically move log rows to a historic database for data mining and purge them in the transactional database. It is a common practice.

I do not think there's the official limit, it'll rely on maximum index dimensions and filesystem limitations.

From mySQL 5.0 Features:

Support for big databases. We use MySQL Server with databases which contain 50 million records. We are conscious of customers who use MySQL Server with 200,000 tables contributing to 5,000,000,000 rows.

There's most likely some kind of limitation, determined by the engine used and also the table structure. I have got a table with appx 45 million records inside a database I administrate, I have heard about (much) greater amounts.