Yes, the 3K rows limit is regardless of your Eikon account. It is aimed to protect the backend infrastructure. If you need more that 3K rows of price history you need to execute multiple get_timeseries methods.
To retrieve monthly timeseries use interval='monthly' keyword argument, e.g.

ek.get_timeseries('TRI.N', 'CLOSE',start_date='2013-01-02', end_date='2018-07-02',interval='monthly')

For full list of parameters you can use with get_timeseries method see Eikon Data APIs for Python Reference Guide available under the Documentation tab on Eikon Data APIs page on this portal.

I think you're likely hitting the 3K limit on the number of rows returned by any single get_timeseries call for any interday interval. This limit is shared by all RICs in the request. Here are the links to a couple of previous threads on the subject.
https://community.developers.refinitiv.com/questions/22511/max-rics-for-get-timeseries-request.html

https://community.developers.refinitiv.com/questions/21902/python-amount-of-datapoints-allowed-per-request.html

Thx Alex. So this is a limitation (3k rows) that will exist regardless of the account?

Also, how can I modify my ek.get_timeseries to pull monthly closes instead of daily?