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At 3645 m, the (reportedly) highest European mountain accessible by regular hiking (no glacier gear) is Monte Vioz (https://en.wikipedia.org/wiki/Monte_Vioz) in Trentino, Italy. The climbing season streches from June to September.
Another similar mountain, though not the highest one available, is Hochfeiler / Gran Pilastro, 3510 m, in South Tyrol, Italy, on the border with Austria, but accessible only from the Italian side. Unlike Monte Vioz, I've already climbed Hochfeiler and it was a spectacular adventure, highly recommended.
Upvote:4
If you consider regular hiking only (without glacier crossings), one contender would be Rocciamelone at 3,538 m.
Warning: although the hike itself is not particularly dangerous, weather can be deadly any time of year at this height, in particular if you do not bring proper clothing. Don't go here in poor weather and bring warm clothes no matter how nice the weather looks.
Unlike higher options, you do not need to cross any glaciers. It's a regular hike and it's not even particularly difficult, mostly T2 with at worst some sections of T3. It's just a long way up and you might notice the reduced oxygen levels up there, so take your time. There are cabins on the way up where you can sleep and there's a small primitive shelter at the summit as well, so you can be safe on the summit if the weather turns bad (or even sleep there if you bring some sleeping and cooking gear, but I'm not sure if it's normally allowed to plan to sleep there; it wasn't when I was there, but I don't know if that was regular or due to corona). You can also drive a bit of the way up, but it remains a long hike even from the highest point one can drive (and I think parking is limited). There are no cable-cars and there is no summit restaurant, so crowds remain reasonable.
I think highest including "easy mountaineering" is hard to define, because easy is subjective, but Rocciamelone might be one of the highest without any mountaineering.
It would be an understatement to say that the sunrise is most definitely worth it. Just can just-about-but-not-quite see the summit of the Mont Blanc:
Rocciamelone sunrise. With Les Bans, Mont Pelvoux, Pic Coolidge, Barre des Γcrins, Grande Ruine, La Meije with Pic Gaspard and Doigt de Dieu, 2021-09-17.
Rocciamelone sunrise with frozen Maria, 2021-09-17.
Rocciamelone sunrise with Massif du Mont Blanc and Grandes Jorasses, Grand Combin, Gran Paradiso, Weisshorn, Matterhorn, Monte Rosa, 2021-09-17.
Rocciamelone view of Glacier de Rochemelon, with La Grande Casse, Grand Motte, Mont Pourri, Massif du Mont Blanc (behind Pointe de Charbonnel), Grande Jorasses, Mont Dolent, l'Albaron, 2021-09-17.
With some luck, you might see a Brocken Spectre and glories at night: