West Bengal shares boundaries with three countries: Bangladesh, Bhutan, and Nepal.

1. Bangladesh: West Bengal shares its longest international border with Bangladesh, spanning around 2,216 kilometers. The border is formed by the states of West Bengal, Tripura, Assam, and Meghalaya. The border with Bangladesh is highly significant for trade, cultural exchange, and migration between the two countries. Several major border crossings are located in West Bengal, including Petrapole-Benapole, the largest land port in South Asia.

2. Bhutan: West Bengal shares a small border with Bhutan in the northern part of the state. The border is approximately 183 kilometers long and is primarily mountainous and forested. The Indian state of West Bengal shares its border with Bhutan's districts of Chukha, Samtse, and Alipurduar. The border with Bhutan is important for trade and tourism activities.

3. Nepal: West Bengal shares a short border with Nepal in its northwestern region. The border is about 98 kilometers long and is formed by the Indian state of West Bengal and Nepal's eastern districts of Jhapa, Ilam, and Panchthar. The border with Nepal facilitates cross-border trade and cultural exchange, with several border checkpoints allowing movement between the two countries.

In addition to these three countries, West Bengal is also bordered by the Indian states of Sikkim, Assam, Jharkhand, and Bihar. However, the question specifically asks about the number of countries that share boundaries with West Bengal, hence the correct answer is three.