Does As Foretold let you cast spells with additional costs like kicker for free?

As Foretold (and similar effects) grant you permission to cast a spell "without paying its mana cost" (CR 601.2b). This replaces only the spell's mana cost — the cost printed in the top-right corner of the card — with (0). It does not remove or waive any additional costs.

Optional additional costs such as Kicker, Entwine, or Buyback are defined in CR 702.32 (Kicker) and CR 601.2b as costs you may choose to pay on top of the base cost. If you choose to pay an optional additional cost, you must still pay it even when casting for free via As Foretold. The free-casting permission only covers the mana cost itself.

Mandatory additional costs (e.g., sacrificing a creature as required by a spell like Fling) must always be paid regardless — As Foretold does not remove those either (CR 601.2f).

Concrete example: You have 3 time counters on As Foretold and cast Grunn, the Lonely King (CMC 5) for free using As Foretold. You want to pay the Kicker cost of (5) so Grunn enters with +5/+5 counters. You pay (0) for the mana cost (covered by As Foretold) but must still pay (5) for Kicker separately. Total out-of-pocket cost: (5).

So the answer depends on what you mean by "for free": the base mana cost is covered, but any kicker or other additional costs you opt into are still your responsibility to pay.