剧院里的人数四舍五入到最接近的百位是 3400。

1. Determine the least possible number that rounds to 3400 to the nearest hundred

When rounding to the nearest hundred, the number must be at least $3400 - 50 = 3350$. Any number from $3350$ to $3399$ rounds up to $3400$. Therefore, the least possible number is $3350$.

1. Determine the upper bound for rounding to the nearest hundred

When rounding to the nearest hundred, numbers from $3350$ up to $3449$ round to $3400$.

2. Identify the greatest possible integer

Since the number of people must be a whole number, the greatest possible value is $3449$.