Learn German: Ich habe den Schmied am Amboss hämmern sehen. - I saw the blacksmith hammering at the anvil.

This is an example of the double infinitive rule (often called the Ersatzinfinitiv). When perception verbs like sehen or hören are used with a second infinitive (like hämmern) in the perfect tense, they drop their standard past participle (gesehen) and use their infinitive form instead.

Why doesn't hämmern go at the very end of the sentence?

In a double infinitive construction, the standard word order rules change slightly. The dependent verb (hämmern) comes first, and the governing verb that would normally be a participle (sehen) must take the absolute final position in the clause.

Why do we use am for 'at the anvil' instead of bei?

The preposition an (which merges with the dative dem to form am) is used here because it describes being positioned right up against a workspace or surface. Beim Amboss would just mean the blacksmith is somewhere in the general vicinity of it.

Could I say 'Ich sah den Schmied am Amboss hämmern' instead?

Absolutely. That uses the simple past (Präteritum) instead of the perfect tense. In fact, many native speakers prefer the simple past for verbs like sehen and hören specifically because it neatly avoids the double infinitive construction entirely!