An additive white Gaussian noise (AWGN) energy-harvesting (EH) channel is considered where the transmitter is equipped with an infinite-sized battery which stores energy harvested from the environment. The energy arrival process is modeled as a sequence of independent and identically distributed (i.i.d.) random variables. The capacity of this channel is known and is achievable by the so-called best-effort and save-and-transmit schemes. This paper investigates the best-effort scheme in the finite blocklength regime and establishes the first nonasymptotic achievable rate for it. The first-order term of the nonasymptotic achievable rate equals the capacity, and the second-order term is proportional to -sqrt log n/n - where n denotes the blocklength. The proof technique involves analyzing the escape probability of a Markov process. In addition, we use this new proof technique to analyze the save-and-transmit and obtain a new non-asymptotic achievable rate for it, whose first-order and second-order terms achieve the capacity and the scaling -1/sqrt n respectively. For all sufficiently large signal-to-noise ratios (SNRs), our new achievable rate outperforms the existing ones.