Hit or Hold at the Indifference Point

Aahna's balance starts at $U_1 \sim$ Uniform(0,1). At each turn she may hold, keeping her balance as her payout, or hit: draw a fresh Uniform(0,1) and add it to the balance. If the balance ever exceeds 1 she gets nothing. Optimal play stops the first time the balance exceeds a threshold $t$.

Find $t$.

Answer format: round to 3 decimal places.