In the sentence <em>We were hoping to begin a game of chess</em>, <em>to begin</em> is an infinitve functioning as a noun.
An infinitive, <em>to </em>plus a verb form, can be used as a noun, adjective or adverb. In this case, as it is functioning a direct object, is functioning as a noun. <em>To begin a game of chess</em> is the object of the the verb <em>hope</em>, is the answer to the question "what were we hoping?"
