I believe you can use a cork borer and remove a core from the tree. Then you may count rings and have more insights on the oldest rings. However, it depends on the diameter of the trunk. You should probably look for a very long cork borer.
Of course this is an invasive technique but it's less dramatic than cutting the tree down. What is the purpose of your study?
yes a cork borer or drill is the best way to obtain a sample for counting rings. Remember though that it may not give a definite age as in some years trees may produce 2 rings, depending on seasonal conditions.
After drilling and removal of the core, you should plug the hole with an inert material to prevent disease.
This link gives an introduction to tree coring and preparing the core for analysis: http://www.ohio.edu/people/dyer/Tree_Core/
As suggested by other colleagues, coring is the best option to save the precious Tree. However, immediately after coring the cavity created needs to be sealed with antibiotic material to prevent any infection.
Alessio, Sally and Syed, many thanks for your sound reply. To answer Alessio, this venerable tree is still producing, it stands in Southern Italy, and I had commented the person who sent me a picture of it, a comment saying that this tree can tell how it survived the Medieval warming, the Renaissance cooling and was enjoying the present wealth of CO2. I was then asked how to procede to get access to the data.
coring is the best solution for your purpose, but be aware of the fact that tree rings in olive trees are difficult (sometimes impossible) to detect even on a complete trunk section. To get an idea I can suggest to read the paper by Paolo Cherubini et alii The olive-branch dating of the Santorini eruption, published on Antiquity feb 2014.
Dear Nicoletta, thanks for this interesting addition. Santorini (Terra) is a fascinating subject. There were at least 5 eruptions during the last 200000 years;is it the one that triiggered the exode of the Hyksos in 1350 BCE or the preceeding on in 1550 BCE that coincides with Akhenaton? Will be thrilled to learn more.