I'm going to use for this pre-Hanover because Georges I, II, III were essentially constitutional monarchs. Just my own view because a constitutional monarch doesn't have much power:
- Alfred the Great
- Aethelstan
- William I
- Edward I/Longshanks
- Henry VII
- Henry VIII
- Elizabeth I
- James I of England/James VI of Scotland
- William III
- Anne I
IMO they all had the most influence. I didn't include King John because whilst he signed Magna Carta, it only was because he lost the Baron War and was forced to ensure he wouldn't violate nobles' rights again. And he lost Normandy to France, which was more or less his family's homeland.