Yes, 3 dogs are a pack and you will start to see behaviours that you did not see when it was just 2 dogs.
When it was just me, Lugia (Silken Windhound) and Circus (Australian Koolie) it was very easy. There were never any fights or conflicts, I never had to do much managing when it came to resources (whether it was myself, food, toys, even highly valued things like RMBs) they ALWAYS got along. The both of them also did very well whenever they interacted with other dogs (at the dog parks, play dates, etc).
About a year ago I moved in with my SO who already had a dog (then, a 1 year old male neutered Lab mix). It took some time for our dogs to get used to living with each other. Lugia (an angel) never caused any problems during this transition. Circus and Pablo would frequently get into it - it seemed like the both of them were resource guarding various things - whether it was us, toys, food, chews, whatever. Neither my SO and I were very experienced in managing a multi-dog household, but we quickly had to learn. We both were used to giving RMBs to our own respective dogs when we lived apart and just forgetting about it and doing our own thing - not the case now, they must be strictly supervised. I used to leave a toy bin out for Lugia and Circus - can't do that anymore cuz Pablo and Circus might get into an altercation over a toy. When it comes to giving attention to any of our dogs, it's now important to be very aware and alert as to the whereabouts and emotional status of all the other dogs in the room - if two dogs are trying to sit in one person's lap and they get too excited, that might erupt in a fight, too. With proper management, we can avoid that kind of conflict... but it really takes diligence.
Ever since Circus turned 2, she stopped being so "DOGGY". When she was younger, she used to be obsessed with interacting with other dogs and very playful, friendly, and submissive to even strange dogs. But now, as she's maturing, I'm noticing a lot more reactivity from her towards other dogs. It's completely manageable IF I'm on top of my game, but I find that I have to do a lot more micro-managing when Circus is interacting with other dogs. She does NOT like it when other dogs are rude and is very ready and willing to take their face off at the first offence it seems... the other day at a feed store (both dogs were offleash) a male castrated 90lb GSD stood over her and kind of stared at her... within a second, she was trying to fight him, even though she's a little peanut at 25 lbs.
She's a feisty one but I'm really trying to work on her reactivity towards other dogs. I wasn't expecting this because I read that Koolies were great with other dogs... Circus WAS! She was perfect with other strange dogs. Uuuuppp until she hit two years old. Now she's a work in progress again. She really is improving and I'm happy with her progress, BUT it is taking a lot of work and training.