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If you are measuring AC then you could consider a Current Transformer - take care as the secondary winding of these things can kick out amazing voltages if open-circuited.
For DC I know of no alternative to shunts - if you have a car battery tested at a garage they will use a gizmo with a huge shunt to measure short-circuit ability. It looks like an inch-wide, two foot long strip of steel for the shunt.
Could you use a hall-effect transistor strapped onto the side of a conductor and infer amps from magnetic field?