Let $a_0, a_1, a_2, \ldots$ be an increasing sequence of nonnegative integers such that every nonnegative integer can be uniquely represented in the form $a_i + 2a_j + 4a_k$, where $i,j,$ and $k$ are not necessarily distinct. We aim to determine $a_{1998}$.

The uniqueness condition suggests that the sequence of $a$'s behaves similarly to a positional numeral system. Specifically, each $a_i$ acts like a digit in a base-8 (octal) system due to the coefficients $1, 2,$ and $4$, which suggest powers of 2.

To represent any number $N$ uniquely as $a_i + 2a_j + 4a_k$, each index $i, j, k$ corresponds to a digit in base-8 representation, i.e., $i, j, k$ select which terms $a_n$ represent the "digit" places in the expansion.

Thus, the numbers $N$ can be expanded in a form similar to a base-8 system where each digit spans from 0 to the maximum allowable index. This insight guides us to choose each $a_i = i$, which aligns the sequence of $a$'s directly with the indices required for base expansion.

Considering the sequence as $a_n = n$, we have:

- $a_0 = 0$,
- $a_1 = 1$,
- $a_2 = 2$,
- $a_3 = 3$,
- ...
- $a_n = n$.

For $a_{1998}$, we recognize $1998$ as a straightforward positional representation in base 8. Thus, converting 1998 from decimal to base 8, we obtain:

$$1998_{10} = 11111001110_2$$

Converting this binary to octal (since every three binary digits correspond to one octal digit):
$$11111001110_2 = 3736_8$$

Therefore, $a_{1998}$ corresponds directly to this octal representation.

Thus, the solution for $a_{1998}$ is given by the base-8 representation:
$$\boxed{3736_8}$$