SnackTime Foods produced 208,000,000 grams of popcorn last year, which was placed in 4-gram bags. How many bags did the company fill?

To find the number of bags the company filled, divide the total weight of popcorn produced by the weight of each bag: 208,000,000 grams / 4 grams/bag = <<208000000/4=52000000>>52,000,000 bags. Answer: \boxed{52,000,000}.