Duplo and Lego: Both Duplo and Lego have created some really incredible free apps for both toddlers and kids and the great thing about them is that they are completely free. Also, all of them have no issues with advertising and in-game purchases which are the bane of many parents.
Train, is a duplo app for toddlers that allows them to pick the train and carriages, load them and unload them. They also have to solve puzzles such as building a bridge and building a track.
Safari, is another duplo app for toddlers which involves a giraffe and rabbit trying to deliver a package to a lion. Hillarious I know, but the little ones have a great time playing with the animals and solving problems to get the package to the lion.
Junior's Quest, is a lego app about a police officer trying to catch a bad guy. The little ones have to build things to help solve problems, drive a car around obstacles (or just watch the car smash through the lego blocks with no penalty) and eventually catch the bad guy. Some of the puzzles are a little hard at first for toddlers, but with a little help that can get the hang of solving them.
Train Conductor: This is a great app for both adults, kids and toddlers. The aim of the game is to direct the right train on the right track so that it can get to its destination. For adults and older kids this game provides the enjoyment of an arcade game, for the younger ones it allows them to have fun either crashing the trains or avoiding crashes. There is some in game advertising on the free version, and in game purchases also. This app may require adult supervision.
Monster Truck Mike: This is a great little app for boys. It's designed for older kids and teaches them the ins and outs of repairing an engine on a monster truck, the types of tools used and then lets them paint it and send it out to a monster truck rally. For younger children supervision is needed. It has in game advertising and purchases, but is definitely worth it.
Cars: This is the Disney cars app game. It can be entertaining for toddlers all the way to adults. It has a small townville portion to the game where you build up radiator springs and place building and decorations. It also has a racing portion that is simple enough that a toddler can play. No in game advertising, some in game purchases.
Cars match games for toddlers: This is a great little matching game app. It places a whole range of vehicles floating around on the page. You child has to grab the vehicle with their finger and drop it on top of another vehicle just like it. The vehicles make noises when touched and when a level is completed your child gets to pop balloons that appear on the page. Some in game ads.
kids trains: This is a very basic train app. A train travels around the track and your little one can control the speed and honk the horn. They can change the train and carriages. They can also crash it, often leading to a world of giggles.
Toddler tools: This is a very basic button pushing app for toddlers and babies. They can press on a picture and it will tell them what type of tool it is and play some music. In game advertising.
Top truck: This is a monster truck driving game designed for toddlers up to adults. The goal of the game is to drive a monster truck forward and back to over come various obstacles put in front of them. The tablet sometimes needs to be rotated to rock the monster truck to help them get over an obstacle. While you wouldn't think this was a good app for toddlers, they pick it up pretty quickly. Some in game advertising.