Can I patch up a straw hat? I have an old straw hat that suits me really well. However, after years of use the front of the crown is worn out. It's a little hard to see in the photo below, but a hole has developed that I could fit my finger through. Is there some way in which a worn out part of a straw hat can be repaired or patched up, or is this a lost cause?

It would be preferable to restore the original appearance if possible, but I'd be OK with anything that looks neat. (I previously tried stitching it up with straw-coloured string, but the stitches needed to be quite big and I couldn't get them even, so it didn't look good at all.)
Your hat has a very open structure, so, to fix a hole invisibly is hard/impossible, as you need to weave in material where it's missing, but then you got a double amount of material at the edge of the hole.
However, since you 'just' want to make it look nice, you could


reweave the hat (with either straw, wool, or any other material of thead) about as thick as the original material. It's obviously a lot of work and you loose the open structure. But you don't have to reweave the complete hat, you could skip the rim, or any other parts without holes.
Stitch/embroider a ribbon along the edge of the crown, using whip stitch (If you don't know this stitch, see https://nl.ifixit.com/Guide/How+to+Sew+a+Whip+Stitch/21025) Make the stitches large, and be sure to cover the hole with the ribbon.
Stick a flower in the hole, and attach it in a creative way. :-)