Mirror, Mirror On the Wall
One of Bug's favorite games early on (as young as 6 weeks) was to look into a mirror at herself. You'll need to hold the hand mirror for your little or hold your little up to a mirror, but it's a great time for littles to look into a mirror. It can also be a good learning tool for identifying body parts, identifying people and so on. You can point to baby and say "baby" and they will see the baby (themselves) in the mirror.
Bug's favorite activity was bath time. We didn't soap her up every night, but once your infant is about 5 or 6 months, I see no reason why he or she can't play in the tub every night. No need to buy special bathtub toys if you don't or can't spend the money: measuring cups, regular cups, wash cloths, etc. are all game for toys for baby.
Bubbles can be a great thing for baby to see. They're so fun and don't require a lot of activity on baby's part beyond just seeing and feeling. Bug loved bubbles (and still does).
If you have a high chair, you have the perfect set-up for sensory play with your infant. We put Bug in her high chair and put flour or water on her high chair tray. That was good fun for her to play with.
Bug obviously cannot read, but that doesn't stop her from loving to flip pages in a book. Before she was 1, we did expose her to real books so she could start to learn to turn paper pages. She is really good at it, but I recommend starting with board books. As your baby progresses with board books, slowly, gently and patiently introduce real books. You'd be surprised how gentle your baby will learn to be with books with a bit of guidance and a lot of practice.
Smile for the Camera
My girl's already outgrown this activity for now, but she was quite obsessed with the camera early on. I would bring it in close so she could see it, take it out further and encourage her to play and be silly for the camera while I took photos. She really enjoyed doing that for a long time, but now she's over the camera. Sniffle.
Sing Your Heart Out
I'm a terribly awful singer, but Bug has loved music since she was in my womb. Even in the car, if she's a little fussy, all we have to do is turn on worship music and she settles right down. It's magic. She has heard worship music since she was 1 month in my belly, and she loves us to sing to her. When she was just a wee one, she would be so transfixed. Now she tries to sing along with us. It doesn't really matter so much what you sing as long as you're confident about it. I stick with worship songs or Christmas songs because I know the lyrics and music. Sing what you know and be happy while singing. That's all your baby really cares about.
Kids love high-contrast ideas. Playing cards are perfect for them. I would just take a deck of cards and slowly show each card to an infant Bug. She was so mesmerized by them.
Touch and feel board books are great for kids because they love to touch new textures. Another idea you can do is create your own textured cards: gather up a bunch of textured items and make textured cards. Think sandpaper, textured cardstock from a crafts store, a painted surface, etc.