Now the first really spurious claims are from 1066 when Saxon Edgar Aetheling was pronounced king instead of William the Conqueror. There was a massive hunt for his descendants last year , and one could surmse that it is tenable that across the globe a direct descendant exists ; but this was all really resolved when Saxon and Norman lines were united in the person of Henry II , grandson of Henry I and his wife Matilda, daughter of Queen Margaret of Scotland who was the daughter of Edward the Exile, son of Edmund Ironside, who was the son of Aethelred the Unready and elder brother of Edward the Confessor.
Next comes the claim that Edward Earl of March was not the legitimate son of Richard , 3rd Duke of York ; and even if he was, his marriage to Elizabeth Woodville was bigamous. In either case the succession fell to his brother George, Duke of Clarence [yes the Malmsey wine guy]
Edward III of England
Lionel of Antwerp, 1st Duke of Clarence, third son of Edward III
Philippa Plantagenet, 5th Countess of Ulster, only child of Lionel
Roger Mortimer, 4th Earl of March, first son of Philippa
Anne de Mortimer, first daughter, third line of Roger
Richard Plantagenet, 3rd Duke of York, only son of Anne
George Plantagenet, 1st Duke of Clarence, third son (second "legitimate" son) of Richard
Margaret Pole, 8th Countess of Salisbury, second daughter, fourth line of George
Henry Pole, 1st Baron Montagu, first son of Margaret
Catherine Hastings, first daughter, second line of Henry
George Hastings, 4th Earl of Huntingdon, second son of Catherine
Francis Hastings, only son of George
Henry Hastings, 5th Earl of Huntingdon, only son of Francis
Ferdinando Hastings, 6th Earl of Huntingdon, oldest son of Henry
Theophilus Hastings, 7th Earl of Huntingdon, only son of Ferdinando
Theophilus Hastings, 9th Earl of Huntingdon, second son of Theophilus
Elizabeth Rawdon, 16th Baroness Botreaux, only daughter, second line of Theophilus, 9th Earl
Francis Rawdon-Hastings, 1st Marquess of Hastings, first son of Elizabeth
George Rawdon-Hastings, 2nd Marquess of Hastings, eldest legitimate son of Francis
Edith Rawdon-Hastings, 10th Countess of Loudoun, first daughter, third line of George
Paulyn Francis Cuthbert Rawdon-Hastings, second son of Edith
Edith Maud Abney-Hastings, 12th Countess of Loudoun, first daughter, third line of Paulyn
Barbara Abney-Hastings, 13th Countess of Loudoun, first daughter, second line of Edith
Michael Abney-Hastings, 14th Earl of Loudoun, eldest son of Barbara
Note: This list excludes females from the crown. The precedent for female inheritance of the Crown would not have been set had George, Duke of Clarence inherited the crown. The principle that a woman could reign was only laid down by Henry VIII when he named Mary I as heir to the throne in 1525, and approved by Parliament by the First Succession Act which appointed Elizabeth I as heir. This line does, however, maintain the precedent of the right of a male to inherit via female line set by the succession of Henry II after he reclaimed the usurped crown from his cousin Stephen.
Now James I only had a claim to the throne via his descendency from Henry VII via his daughter [James's great-Grandmother [paternal and maternal] ] Margaret Tudor - but the Henry VIII's Third act of succession decreed the line of succession had to pass Margaret's younger sister Mary.
Had Elizabeth I upheld this and not decreed James I [VI] her successor; our present Queen would be Lady Caroline Ogilvy.