He is the 19-year-old son of Jussi Jaaskelainen, who made 529 appearances in 15 seasons at Bolton, also playing for West Ham and Wigan Athletic.
The teenager was released by Bolton at the end of last season without having made a competitive appearance.
Crewe have started the new season with draws at home to Mansfield and Newport and a defeat by Bolton in the EFL Cup.
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Crewe Alexandra have signed Finnish goalkeeper Will Jaaskelainen on a one-year-deal after leaving Bolton.