The .308 Win (.308" bullets, same as the .30-06) is the 'get anything done' round. I've used it a lot on almost everything. While very utilitarian, a significant factor in my having used it so much has also been logistics. Since I live in the USA, I can go into any Walmart in pretty much any town in America and get however much .308 Win I need. This round is really just a shortened .30-06 that fits a short action. In rifle matches, due to the more efficient and uniform burn of the powder in the shorter case, it tends to out-perform the .30-06 in accuracy.
One of my personal favorites has always been the 6.5x55mm Swede round. In the 1990's when Sweden cleared out their reserve inventories of 6.5x55mm weapons to put their 7.62x51mm HK G3 rifles into reserve (they had recently adopted a new 5.56mm military rifle that was a modified FNC) the US got hit with a wave of cheap, very old but excellent condition, Swedish made Mauser 96 rifles in 6.5x55mm and a large amount of ammo. The cheapest I ever paid for one was $60. Many of us got in a lot of trigger time with the 6.5x55mm and a huge number of military rifles got sporterized. Just about every one of those rifles was an absolute tack driver.
You can literally kill anything this side of a T-Rex with the 6.5x55mm, the recoil won't knock the fillings out of your teeth, and 6.5mm bullets are in a ballistic sweet spot with sectional densities and ballistic coefficients which are superb. Ammunition and reloading components for it should be available in Europe (Norma brand especially), except France. The last I checked, France had some odd laws that went back almost to the dawn of the Industrial Age on what constitutes a 'military caliber', which you need special licensing for.
It's ballistic step brother (up to 140gr bullets) is the .260 Remington, which is essentially a .308 round with a 6.5mm bullet (.264") in it. It's big advantage is that it will fit into a short action and it's never been a military round. The .260 Rem can be described as the kick of a .243, the power of a .308, and the trajectory of a .300 WinMag. You can make the brass from 7mm-08 or .243 cases. You can also make it from .308 Win or military surplus 7.62x51mm NATO cases but you have to typically trim the excess thickness of the brass at the case mouth when it's fully necked down.
A close runner up to the .260 Rem is the 7mm-08. It's a .308 necked down to take a 7mm bullet (.284"). You can manufacture your own cases out of surplus military 7.62x51mm NATO brass. It uses the same diameter bullets as the 7x57mm Mauser round. You will find ammo and components a lot more available for the 7mm-08 than the .260 Rem.
Another couple of personal favorites are the .25-06 and the .270 Winchester. These are .30-06 cases that use .257" and .277" bullets.