This experience comes from my first area in the town of Bella Vista, Corrientes. I was with my second companion Elder Bullock (we also happened to have graduated from the same high school). One day while out tracting we ran into a man who did not seem interested in talking to us at that time but was very enthusiastic about us coming back later that evening. Because we didn't have anything else that evening we agreed.
A few hours later we arrived at his house and he was still eager to talk to us, and demonstrated his interest by bringing out a few books to use in the discussion. In retrospect I don't think any first discussion I had on my mission ever went well when the person was excited to talk to us. The discussion started like just about any other discussion. We talked about the Church and Joseph Smith. At one point we read a scripture from the Bible and the guy asked if he could see my companion's Bible. As missionaries we were aware of the common criticism against the Church that we use a different Bible (we don't, at the time we used the Reina-Valera 1960 version) so my companion handed over his Bible so the man could see for himself that we have a standard Bible.
The man quickly took the Bible, opened his own and began to compare the title page of his Bible to ours. He began counting the number of books and because he was comparing our Protestant derived Bibles to his Catholic Bible he came up with a different number of books (seven to be exact). Whereupon he handed back my companion's Bible and declared that it was wrong because it was incomplete. To which my companion replied, "Oh really? Do you know which books my Bible is missing?"
"No." he replied.
"Do you know what is in the books that your Catholic Bible has that mine doesn't?"
"No." again was the reply.
"Then how do you know that our Bibles are wrong?"
"But your Bibles don't have the correct number of books!"
"But if you don't know what is in those extra seven books then how do you know that we are actually leaving out something important?"
Realizing that this line of attack wasn't getting him anywhere and that it was actually demonstrating his ignorance rather than ours he switched tactics. He changed the topic by bringing up polygamy. He said that we could not be Christian since we practiced polygamy. My companion assured him that we do not currently practice polygamy. The man insisted that we still do practice polygamy.
Intrigued my companion asked where he was getting his information. At this point he pulled out a little book from his stack of books. Essentially it was a primer on different religions written from a Catholic perspective. The section on Mormons was, shall we say, severely lacking in logic and relevant factual information. He read us a few paragraphs from his book and in that selection he read a quote that essentially stated something to the effect, "There are X number of polygamists living in Utah. Of the people in the state Y% are Mormons. Therefore Mormons still practice polygamy."
Even at the tender age of 19, without ever taking a class on logic, I knew this was bad logic. I pointed this out to him and he became defensive. My companion, a bit amused at his lack of logical rigor, asked where his book got the quote from. He said something that neither one of us could understand. He read the source again from the book but we couldn't make out what he was saying. I asked to see the book and he handed it over. I found the quote and read the source. It was from the Ladies Home Journal, from the 1920's or 30's. At this point my companion laughed out loud much to the confusion of the man we were talking to. My companion had to explain what the Ladies Home Journal was to him (I think my companion called it "Chisme de las mujeras." "Ladies gossip."). Our adversary was beginning to realize that he probably hadn't picked the best resource for a gospel fight. He was about to learn that he was seriously spiritually out gunned.
We went back and forth for a bit but we realized that we weren't getting anywhere. He offered a few more arguments, but eventually his arguments devolved into general objections against religion and not just our religion in particular. He was also realizing that he actually didn't have any good arguments.
It was at this point that my companion commented and said, "It seems like you have a lot of doubts."
He hung his head slightly and admitted that there was a lot of things he had doubts about.
What followed was one of the most powerful testimonies I ever had the opportunity to hear on my mission. My companion leaned forward and said, "I know that what I believe is true. I know that what we have in the scriptures is the word of God. I have no doubt in my mind that Joseph Smith was a prophet of God and that he translated the Book of Mormon. There is no doubt in my mind about God and His Church on the earth. You may doubt but I have no doubts about what I know and your doubts and what you don't know can never change what I do know and what I don't doubt."
The man, struck by the forcefulness of my companion's testimony compared to the weakness of his own doubts quickly shot back and threw up a defensive wall of, "No! I actually do know things!" and "I don't have any doubts." But it was to no avail we both had heard him say in all honesty and sincerity that he was full of doubts. We wished him a good night and left.