I have one that only requires hiking boots during the entire climbing season and slightly beats the Barrhorn: Monte Vioz in Italy, on the border of the Lombardy and Trento regions, at 3645 meters.
The standard route, SAT number 105, is from the village of Pejo on the Trento side, and is a simple mountain hike. There are a couple of places with fixed ropes but even they are not needed. It is certainly a lot of altitude gain, over 2000m, and if you're not used to >3000 meter altitudes, it might make you feel dizzy, but as long as that is something you can deal with you should be good. There might be a small amount of snow at the summit in early summer, but nowhere near enough to require crampons. Note that the approach from any other direction is more challenging and involves glaciers.
There is also an alternative route, 105A, which joins the main route at around 2900 meters. This is connected to a cable car going up to nearly 3000 meters as well as to Rifugio Doss dei Cembri at around 2300 meters, from where there is a wide network of easy footpaths and gravel roads. This route is actually slightly trickier, with lots of loose rocks. Also nothing you need equipment for, but I saw lots of families with small children trying to walk through it (presumably since the altitude difference is much smaller if you take the cable car), and that seems somewhat inadvisable to me. I don't know if this alternative route used to be simpler, as there have been rockslides here.