Arthur's Seat is the strangest thing about Edinburgh: a 350-million-year-old volcano left standing in the middle of the city, wild enough to feel like the Highlands, ten minutes from the castle. You don't tame it. You walk up it and let it show you everything.
About five and a half kilometres, and a proper hill — give it the morning. From the park gate by the Parliament, head for the Salisbury Crags and walk the long dolerite curve, the Radical Road, with the Old Town falling away beneath the cliff. The path tilts up toward the summit from the south; take it steadily. The last few metres are a scramble over bare basalt, and then the whole world opens — the Forth, the bridges, the Pentlands, the sea, the castle small below you.
Come down the eastern flank past Dunsapie Loch, where the swans don't care about the view, then over to the lone gable of St Anthony's Chapel above St Margaret's Loch. Follow the road back under the crags to the Parliament, legs earned, and walk five minutes uphill for the best coffee on the Royal Mile.