There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes ...,X_-1,X_0,X_1,... whose partial sums S_n=X_1+...+X_n are of the form S_n=M_n+R_n, where M_n is a square integrable martingale with stationary increments and R_n is a remainder term for which E(R_n^2)=o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting · denote the norm in L^2(P), a sufficient condition for the partial sums of

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... a stationary process to have the form S_n=M_n+R_n is that n^-3/2 E(S_n|X_0,X_-1,...) be summable. A sufficient condition for the LIL is only slightly stronger, requiring n^-3/2^3/2(n) E(S_n|X_0,X_-1,...) to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.