Let $\tau _{D}(Z) $ be the first exit time of iterated Brownian motion from a domain $D \subset \mathbb{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau _{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[\tau _{D}(Z) >t]$ over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116 (2006) 905–916], for $z\in D$
 $ \displaystyle \lim_{t\to\infty} t^{-1/2}\exp\left(\frac{3}{2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}\right) P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber$
where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}\left( \psi(z)\int_{D}\psi(y){\rm d}y\right) ^{2}$. Here λD is the first eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$ in D, and ψ is the eigenfunction corresponding to λD. We also study lifetime asymptotics of Brownian-time Brownian motion, $Z^{1}_{t} = z+X(|Y(t)|)$, where Xt and Yt are independent one-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.